Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board chapter 5 - Vectors [Latest edition]

The vector `bar"a"` is directed due north and `|bar"a"|` = 24. The vector `bar"b"` is directed due west and `|bar"b"| = 7`. Find `|bar"a" + bar"b"|`.

In the triangle PQR, `bar"PQ" = bar"2a", bar"QR" = bar"2b"`. The midpoint of PR is M. Find the following vectors in terms of `bar"a"` and `bar"b"`:

(i) `bar"PR"` (ii) `bar"PM"` (iii) `bar"QM"`.

OABCDE is a regular hexagon. The points A and B have position vectors `bar"a"` and `bar"b"` respectively referred to the origin O. Find, in terms of `bar"a"` and `bar"b"` the position vectors of C, D and E.

ABCDEF is a regular hexagon. Show that `bar"AB" + bar"AC" + bar"AD" + bar"AE" + bar"AF" = 6bar"AO"`, where O is the centre of the hexagon.

Check whether the vectors `2hati + 2hatj + 3hatk, - 3hati + 3hatj + 2hatk` and `3hati + 4hatk` form a triangle or not.

In the given figure express `bar"c"` and `bar"d"` in terms of `bar"a"` and `bar"b"`.

Find a vector in the direction of `bara = hati - 2hatj` that has magnitude 7 units.

Find the distance from (4, - 2, 6) to each of the following:
(a) The XY-plane
(b) The YZ-plane
(c) The XZ-plane
(d) The X-axis
(e) The Y-axis
(f) The Z-axis.

Find the coordinates of the point which is located three units behind the YZ-plane, four units to the right of XZ-plane, and five units above the XY-plane.

Find the coordinates of the point which is located in the YZ-plane, one unit to the right of the XZ- plane, and six units above the XY-plane.

Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).

If `bar"AB" = 2hat"i" - 4hat"j" + 7hat"k"` and initial point A(1, 5, 0). Find the terminal point B.

Show that the following points are collinear:

A = (3, 2, –4), B = (9, 8, –10), C = (–2, –3, 1)

Show that the following points are collinear:

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).

If the vectors `2hati - qhatj + 3hatk` and `4hati - 5hatj + 6hatk` are collinear, find q.

Are the four points A(1, −1, 1), B(−1, 1, 1), C(1, 1, 1) and D(2, −3, 4) coplanar? Justify your answer.

Express `- hat"i" - 3hat"j" + 4hat"k"` as the linear combination of the vectors `2hat"i" + hat"j" - 4hat"k", 2hat"i" - hat"j" + 3hat"k"` and `3hat"i" + hat"j" - 2hat"k"`

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hati - hatj + 3hatk`  and `- 5hati + 2hatj - 5hatk` in the ratio 3:2 is internally.

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and  `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.

Find the position vector of midpoint M joining the points L(7, –6, 12) and N(5, 4, –2).

If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find

1. the ratio in which the point C divides the line segment AB
2. the values of p and q.

The position vector of points A and B are `6bar"a" + 2bar"b"` and `bar"a" - 3bar"b"`. If the point C divides AB in the ratio 3 : 2, show that the position vector of C is `3bar"a" - bar"b"`.

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

D and E divides sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of the lengths of the parallel sides.

If two of the vertices of a triangle are A (3, 1, 4) and B(− 4, 5, −3) and the centroid of the triangle is at G (−1, 2, 1), then find the coordinates of the third vertex C of the triangle.

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Find two unit vectors each of which is perpendicular to both `baru` and `barv` where `baru = 2hati + hatj - 2hatk`, `barv = hati + 2hatj - 2hatk`.

If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`

Find the values of c so that for all real x, the vectors `"xc"hat"i" - 6hat"j" + 3hat"k"` and `"x"hat"i" + 2hat"j" + 2"cx"hat"k"` make an obtuse angle.

Show that the sum of the length of projections of `"p"hat"i" + "q"hat"j" + "r"hat"k"` on the coordinate axes, where p = 2, q = 3 and r = 4 is 9.

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = - 9hat"i" + 6hat"j" + 15hat"k"` , `bar"b" = 6hat"i" - 4hat"j" - 10hat"k"`.

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = 2hat"i" + 3hat"j" - hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`

Determine where `bb(bara)` and `bb(barb)` are orthogonal, parallel or neither.

`bara = -3/5hati + 1/2hatj + 1/3hatk ,  barb = 5hati + 4hatj + 3hatk`

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = 4hat"i" - hat"j" + 6hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`

Find the angle P of the triangle whose vertices are P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1).

If `hat"p", hat"q"` and `hat"r"` are unit vectors `hat"p"+hat "r" = hat "q"`, find `hat"p".hat"q".`

If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"r".`

Prove by vector method, that the angle subtended on semicircle is a right angle.

If a vector has direction angles 45° and 60°, find the third direction angle.

If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.

If a line has the direction ratios 4, −12, 18, then find its direction cosines

The direction ratios of `bar"AB"` are −2, 2, 1. If A = (4, 1, 5) and l(AB) = 6 units, find B.

Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.

If `bar"a" = 2hat"i" + 3hat"j" - hat"k"`, `bar"b" = hat"i" - 4hat"j" + 2hat"k"`, find `(bar"a" + bar"b") xx (bar"a" - bar"b")`

Find a unit vector perpendicular to the vectors `hat"j" + 2hat"k"`  and  `hat"i" + hat"j"`.

If `bar"a".bar"b" = sqrt3` and `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`, find the angle between `bar"a"` and `bar"b"`.

If `bar"a" = 2hat"i" + hat"j" - 3hat"k"` and  `bar"b" = hat"i" - 2hat"j" + hat"k"`, find a vector of magnitude 5 perpendicular to both `bar"a"` and `bar"b"`.

Find `bar"u".bar"v"` if `|bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8`

Find `|bar"u" xx bar"v"|` if `|bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12`

Prove that `2(bar"a" - bar"b") xx 2(bar"a" + bar"b") = 8(bar"a" xx bar"b")`

If `bar"a" = hat"i" - 2hat"j" + 3hat"k"`  , `bar"b" = 4hat"i" - 3hat"j" + hat"k"` , `bar"c" = hat"i" - hat"j" + 2hat"k"` verify that `bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"`

Find the area of the parallelogram whose adjacent sides are `bar"a" = 2hat"i" - 2hat"j" + hat"k"` and `bar"b" = hat"i" - 3hat"j" - 3hat"k"`

Show that vector area of a parallelogram ABCD is `1/2 (bar"AC" xx bar"BD")` where AC and BD are its diagonals.

Find the area of parallelogram whose diagonals are determined by the vectors `bar"a" = 3hat"i" - hat"j" - 2hat"k"` and `bar"b" = - hat"i" + 3hat"j" - 3hat"k"`.

If `bar"a", bar"b", bar"c", bar"d"` are four distinct vectors such that `bar"a" xx bar"b" = bar"c" xx bar"d"` and `bar"a" xx bar"c" = bar"b" xx bar"d"` prove that `bar"a" - bar"d"` is parallel to `bar"b" - bar"c"`.

If `bar"a" = hat"i" + hat"j" + hat"k"  "and"  bar"c" = hat"j" - hat"k"`, find `bar"a"` vector `bar"b"` satisfying `bar"a" xx bar"b" = bar"c"  "and"  bar"a".bar"b" = 3`

Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`

If `|bar"a".bar"b"| = |bar"a" xx bar"b"|` and `bar"a".bar"b" < 0`, then find the angle between `bar"a"  "and"  bar"b"`.

Prove, by vector method, that sin (α + β) = sin α . cos β + cos α . sin β

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are - 2, 1, - 1 and - 3, - 4, 1

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2

Prove that the two vectors whose direction cosines are given by relations al  + bm + cn = 0 and fmn  + gnl + hlm = 0 are perpendicular, if `"f"/"a" + "g"/"b" + "h"/"c" = 0`

If A(1, 2, 3) and B(4, 5, 6) are two points, then find the foot of the perpendicular from the point B to the line joining the origin and the point A.

Find `bar"a".(bar"b" xx bar"c")` if `bar"a" = 3hat"i" - hat"j" + 4hat"k" , bar"b" = 2hat"i" + 3hat"j" - hat"k"` and `bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"` 

If the vectors `3hat"i" + 5hat"k", 4hat"i" + 2hat"j" - 3hat"k"` and `3hat"i" + hat"j" + 4hat"k"`  are the coterminus edges of the parallelopiped, then find the volume of the parallelopiped.

If the vectors `- 3hat"i" + 4hat"j" - 2hat"k" , hat"i" + 2hat"k"` and `hat"i" - "p"hat"j"` are coplanar, then find the value of p.

Prove that `[bar"a"  bar"b" + bar"c"  bar"a" + bar"b" + bar"c"] = 0`

Prove that `(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a"  bar"b"  bar"c"]`.

If `bar"c" = 3bar"a" - 2bar"b"`, then prove that `[bar"a"  bar"b"  bar"c"] = 0`.

If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `(bar"u" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`

If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `[bar"u" xx bar"v"     bar"u" xx bar"w"     bar"v" xx bar"w"]`

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

If `bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j"` and `bar"c" = 2hat"i" + hat"j" + 3hat"k"`, then verify that `bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`

If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.

Show that `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"`

Select the correct option from the given alternatives:

If `|bar"a"| = 2, |bar"b"| = 3, |bar"c"| = 4` then `[bar"a" + bar"b"    bar"b" + bar"c"    bar"c" - bar"a"]` is equal to

Select the correct option from the given alternatives:

If `|bar"a"| = 3, |bar"b"| = 4,` then the value of λ for which `bar"a" + lambdabar"b"`, is perpendicular to `bar"a" - lambdabar"b"`, is

If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______.

If `|bara|` = 3, `|barb|` = 5, `|barc|` = 7 and `bara + barb + barc = bar0`, then the angle between `bara` and `barb` is ______.

Select the correct option from the given alternatives:

The volume of tetrahedron whose vectices are (1,-6,10), (-1, -3, 7), (5, -1, λ) and (7, -4, 7) is 11 cu units, then the value of λ is

If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ = ______.

Select the correct option from the given alternatives:

The distance of the point (3, 4, 5) from the Y-axis is ______ 

Select the correct option from the given alternatives:

The line joining the points (2, 1, 8) and (a, b, c) is parallel to the line whose direction ratios are 6, 2, 3. The value of a, b, c are

Select the correct option from the given alternatives:

If cos α, cos β, cos γ are the direction cosines of a line, then the value of sin²α + sin²β + sin²γ  is ______ 

Select the correct option from the given alternatives:

If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______ 

Select the correct option from the given alternatives:

If `|bar"a"| = 3` and - 1 ≤ k ≤ 2, then `|"k"bar"a"|` lies in the interval

Select the correct option from the given alternatives:

Let α, β, γ be distinct real numbers. The points with position vectors `alphahat"i" + betahat"j" + gammahat"k",  betahat"i" + gammahat"j" + alphahat"k",   gammahat"i" + alphahat"j" + betahat"k"`

Select the correct option from the given alternatives:

Let `bar"p"  "and"  bar"q"` be the position vectors of P and Q respectively, with respect to O and `|bar"p"| = "p", |bar"q"| = "q"`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular; then

Select the correct option from the given alternatives:

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC respectively of a Δ ABC. The length of the median through A is

Select the correct option from the given alternatives:

If `bar"a"  "and"  bar"b"` are unit vectors, then what is the angle between `bar"a"` and `bar"b"` for `sqrt3bar"a" - bar"b"` to be a unit vector?

Select the correct option from the given alternatives:

If θ be the angle between any two vectors `bar"a"  "and"  bar"b"` then `|bar"a" . bar"b"| = |bar"a" xx bar"b"|`, when θ is equal to

Select the correct option from the given alternatives:

The value of `hat"i".(hat"j" xx hat"k") + hat"j".(hat"i" xx hat"k") + hat"k".(hat"i" xx hat"j")` is

Select the correct option from the given alternatives:

Let a, b, c be distinct non-negative numbers. If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k" , hat"i" + hat"k"  "and"  "c"hat"i" + "c"hat"j" + "b"hat"k"` lie in a plane, then c is

Let `bara = hati - hatj, barb = hatj - hatk, barc = hatk - hati.` If `bard` is a unit vector such that `bara * bard = 0 = [(barb, barc, bard)]`, then `bard` equals ______.

Select the correct option from the given alternatives:

If `bar"a", bar"b", bar"c"` are non-coplanar unit vectors such that `bar"a"xx (bar"b"xxbar"c") = (bar"b"+bar"c")/sqrt2`, then the angle between `bar"a"  "and"  bar"b"` is 

ABCD is a trapezium with AB parallel to DC and DC = 3AB. M is the midpoint of DC. `bar"AB" = bar"p", bar"BC" = bar"q"`.
Find in terms of `bar"p" and bar"q"`:

(i) `bar"AM"` (ii) `bar"BD"` (iii) `bar "MB"` (iv) `bar"DA"`

The points A, B, C have position vectors `bar"a", bar"b" and bar"c"` respectively. The point P is the midpoint of AB. Find the vector `bar"PC"` in terms of `bar"a", bar"b", bar"c"`.

In a pentagon ABCDE, show that `bar"AB" + bar"AE" + bar"BC" + bar"DC" + bar"ED" = 2bar"AC"`

In a parallelogram ABCD, diagonal vectors are `bar"AC" = 2hat"i" + 3hat"j" + 4hat"k" and bar"BD" = - 6hat"i" + 7hat"j" - 2hat"k"`, then find the adjacent side vectors `bar"AB" and bar"AD"`.

If two sides of a triangle are `hat"i" + 2hat"j" and hat"i" + hat"k"`, find the length of the third side.

If `|bar"a"| = |bar"b"| = 1,  bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find"  |bar"c"|`.

Find the lengths of the sides of the triangle and also determine the type of a triangle:

A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)

Find the lengths of the sides of the triangle and also determine the type of a triangle:

L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).

Find the component form of `bar"a"` if it lies in YZ-plane makes 60° with positive Y-axis and `|bar"a"| = 4`.

Two sides of a parallelogram are `3hat"i" + 4hat"j" - 5hat"k"` and  `-2hat"j" + 7hat"k"`. Find the unit vectors parallel to the diagonals.

If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar"AD" + bar"BE" + bar"CF" = bar0`.

Find the unit vectors that are parallel to the tangent line to the parabola y = x² at the point (2, 4).

Express `hat"i" + 4hat"j" - 4hat"k"` as the linear combination of the vectors `2hat"i" - hat"j" + 3hat"k", hat"i" - 2hat"j" + 4hat"k"` and `- hat"i" + 3hat"j" - 5hat"k"`.

If `bar"OA" = bar"a" and bar"OB" = bar"b",` then show that the vector along the angle bisector of ∠AOB is given by `bar"d" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|).`

The position vectors of three consecutive vertices of a parallelogram are `hat"i" + hat"j" + hat"k", hat"i" + 3hat"j" + 5hat"k"` and `7hat"i" + 9hat"j" + 11hat"k"`. Find the position vector of the fourth vertex.

A point P with position vector `(- 14hat"i" + 39hat"j" + 28hat"k")/5` divides the line joining A (1, 6, 5) and B in the ratio 3 : 2, then find the point B.

ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.

If ABC is a triangle whose orthocentre is P and the circumcentre is Q, prove that `bar"PA" + bar"PB" + bar"PC" = 2bar"PQ".`

If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that `bar"QP" = 3bar"QG"`.

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Dot product of a vector with vectors `3hat"i" - 5hat"k",  2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k"` are respectively -1, 6 and 5. Find the vector.

If `bar"a", bar"b", bar"c"` are unit vectors such that `bar"a" + bar"b" + bar"c" = bar0,` then find the value of `bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".`

If a parallelogram is constructed on the vectors `bar"a" = 3bar"p" - bar"q", bar"b" = bar"p" + 3bar"q" and |bar"p"| = |bar"q"| = 2` and angle between `bar"p" and bar"q"` is `pi/3,` and angle between lengths of the sides is `sqrt7 : sqrt13`.

Express the vector `bar"a" = 5hat"i" - 2hat"j" + 5hat"k"` as a sum of two vectors such that one is parallel to the vector `bar"b" = 3hat"i" + hat"k"` and other is perpendicular to `bar"b"`.

Find two unit vectors each of which makes equal angles with bar"u", bar"v" and bar"w" where bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v" = hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k".

Find the acute angle between the curves at their points of intersection, y = x², y = x³.

Find the direction cosines and direction angles of the vector `2hat"i" + hat"j" + 2hat"k"`

Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively.

Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.

Find the angle between the lines whose direction cosines are given by the equations 6mn - 2nl + 5lm = 0, 3l + m + 5n = 0.

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Show that the vector area of a triangle ABC, the position vectors of whose vertices are `bar"a", bar"b" and bar"c"` is `1/2[bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"]`.

Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b" xx bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a" xx (bar"b".bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a" xx(bar"b" xx bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b".bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b") xx (bar"c".bar"d")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a" xx bar"b").(bar"c"xxbar"d")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b").bar"c"`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b")bar"c"`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`|bar"a"|(bar"b".bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b" + bar"c")`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a". bar"b" + bar"c"`

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`|bar"a"|. (bar"b" + bar"c")`

For any vectors `bar"a", bar"b", bar"c"` show that `(bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a" = 2bar"a" xx bar"c"`

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c"`, then is `bar"b" = bar"c"` ?

Suppose `bar"a" = bar"0"`:

If `bar"a" xx bar"b" = bar"a" xx bar"c"`, then is `bar"b" = bar"c"` ?

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c" and bar"a" xx bar"b" = bar"a" xx bar"c"`,  then is `bar"b" = bar"c"`?

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then verify that the points are the vertices of a parallelogram.

Let A, B, C, D be any four points in space. Prove that `|bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" + bar"BD"|` = 4 (area of triangle ABC).

Let hat"a", hat"b", hat"c" be unit vectors such that hat"a".hat"b" = hat"a".hat"c" = 0 and 6  the angle between hat"b" and hat"c" is pi/6. Prove that hat"a" = +- 2(hat"b" xx hat"c").

Find the value of ‘a’ so that the volume of parallelopiped formed by hat"i" + "a"hat"j" + hat"k", hat"j" + "a"hat"k" and "a"hat"i" + hat"k" becomes minimum.

Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.

If `bar"a", bar"b", bar"c"` are three non-coplanar vectors show that `(bar"a".(bar"b" xx bar"c"))/((bar"c" xx bar"a").bar"b") + (bar"b".(bar"a" xx bar"c"))/((bar"c" xx bar"a").bar"b") = 0`

Prove that `(bar"a" xx bar"b").(bar"c" xx bar"d")` =
`|bar"a".bar"c"    bar"b".bar"c"|`
`|bar"a".bar"d"    bar"b".bar"d"|.`

Find the volume of a parallelopiped whose coterimus edges are represented by the vectors `hat"i" + hat"k", hat"i" + hat"k", hat"i" + hat"j"`. Also find volume of tetrahedron having these coterminus edges.

Using properties of scalar triple product, prove that `[(bar"a" + bar"b",  bar"b" + bar"c",  bar"c" + bar"a")] = 2[(bar"a",  bar"b",  bar"c")]`.

If four points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d")` are coplanar, then show that `[(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`.

If `bar"a", bar"b", bar"c"` are three non-coplanar vectors, then `(bar"a" + bar"b" + bar"c").[(bar"a" + bar"b") xx (bar"a" + bar"c")] = - [bar"a"  bar"b" bar"c"]`

If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair is also perpendicular.