Advertisements
Advertisements
प्रश्न
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".
उत्तर
Let `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k"` be the unit vector which makes angle θ with each of the vectors
Then `|bar"r"| = 1`
Also, `bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v"= hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k"`
`|bar"u"| = sqrt(2^2 + 1^2 + (- 2)^2) = sqrt(4 + 1 + 4) = sqrt9 = 3`
`|bar"v"| = sqrt(1^2 + 2^2 + (- 2)^2) = sqrt(1 + 4 + 4) = sqrt9 = 3`
`|bar"w"| = sqrt(2^2 + (- 2)^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt9 = 3`
Angle between `bar"r" and bar"u"` is θ
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" + hat"j" - 2hat"k"))/(1xx3)`
`= (2"x" + "y" - 2"z")/3` ....(1)
Also, the angle between `bar"r" and bar"v"` and between `bar"r" and bar"w"` is θ.
∴ cos θ = `(bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(hat"i" + 2hat"j" - 2hat"k"))/(1xx3)`
`= ("x" + 2"y" - 2"z")/3` ....(2)
and cos θ `= (bar"r".bar"u")/(|bar"r"||bar"u"|)`
`= (("x"hat"i" + "y"hat"j" + "z"hat"k").(2hat"i" - 2hat"j" + hat"k"))/(1xx3)`
`= (2"x" - 2"y" + "z")/3` ....(3)
From (1) and (2), we get
`(2"x" + "y" - 2"z")/3 = ("x" + 2"y" - 2"z")/3`
∴ 2x + y - 2z = x + 2y - 2z
∴ x = y
From (2) and (3), we get
`("x" + 2"y" - 2"z")/3 = (2"x" - 2"y" + "z")/3`
∴ x + 2y - 2z = 2x - 2y + z
∴ 3y = 3z .....[∵ x = y]
∴ y = z
∴ x = y = z
∴ `bar"r" = "x"hat"i" + "y"hat"j" + "z"hat"k" = "x"hat"i" + "x"hat"j" + "x"hat"k"`
∴ `|bar"r"| = sqrt("x"^2 + "x"^2 + "x"^2) = 1`
∴ `"x"^2 + "x"^2 + "x"^2 = 1`
∴ `3"x"^2 = 1`
∴ `"x"^2 = 1/3`
∴ x = `- 1/sqrt3 `
∴ `bar"r" = +- 1/sqrt3 hat"i" +- 1/sqrt3 hat"j" +- 1/sqrt3hat"k" `
`= +- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
Hence, the required unit vectors are `+- 1/sqrt3 (hat"i" + hat"j" + hat"k")`
APPEARS IN
संबंधित प्रश्न
If \[\vec{a}\] and \[\vec{b}\] represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] represent the sides of a triangle taken in order, then write the value of \[\vec{a} + \vec{b} + \vec{c} .\]
If \[\overrightarrow{a}\], \[\overrightarrow{b}\], \[\overrightarrow{c}\] are the position vectors of the vertices of a triangle, then write the position vector of its centroid.
If \[\overrightarrow{a}\] and \[\overrightarrow{b}\] denote the position vectors of points A and B respectively and C is a point on AB such that 3AC = 2AB, then write the position vector of C.
If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of \[\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} .\]
Write the position vector of a point dividing the line segment joining points A and B with position vectors \[\vec{a}\] and \[\vec{b}\] externally in the ratio 1 : 4, where \[\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\overrightarrow{b} = - \hat{i} + \hat{j} + \hat{k} .\]
If \[\left| \overrightarrow{a} \right| = 4\] and \[- 3 \leq \lambda \leq 2\], then write the range of \[\left| \lambda \vec{a} \right|\].
If OACB is a parallelogram with \[\overrightarrow{OC} = \vec{a}\text{ and }\overrightarrow{AB} = \vec{b} ,\] then \[\overrightarrow{OA} =\]
If \[\vec{a}\text{ and }\vec{b}\] are two collinear vectors, then which of the following are incorrect?
Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line \[\frac{x + 2}{5} = \frac{y - 3}{4} = \frac{z}{5}\] or not.
Find the components along the coordinate axes of the position vector of the following point :
S(4, –3)
Find the position vector of the mid-point of the vector joining the points
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.
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:
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?
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)
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.
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"`.
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.
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" xx 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"`
If the vectors `xhat"i" - 3hat"j" + 7hat"k" and hat"i" + "y"hat"j" - "z"hat"k"` are collinear then the value of `"xy"^2/"z"` is equal.
Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + 2hat"k"` and `vec"b" = -hat"i" + hat"j" + 3hat"k"`.
If `|vec"a"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.
The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC, respectively of a ∆ABC. The length of the median through A is ______.
Find a unit vector in the direction of `vec"PQ"`, where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively
Classify the following as scalar and vector quantity.
Distance
Let `veca, vecb` and `vecc` be three unit vectors such that `veca xx (vecb xx vecc) = sqrt(3)/2 (vecb + vecc)`. If `vecb` is not parallel to `vecc`, then the angle between `veca` and `vecc` is
Check whether the vectors `2hati + 2hatj + 3hatk, - 3hati + 3hatj +2 hatk and 3hati + 4hatk` from a triangle or not.
lf ΔABC is an equilateral triangle and length of each side is “a” units, then the value of `bar(AB)*bar(BC) + bar(BC)*bar(CA) + bar(CA)*bar(AB)` is ______.
Check whether the vectors `2hati+2hatj+3hatk,-3hati+3hatj+2hatk` and `3hati+4hatk` form a triangle or not.
In the triangle PQR, `bar(PQ)` = 2`bara` and `bar(QR)` = 2`barb`. The mid-point of PR is M. Find following vectors in terms of `bara` and `barb`.
- `bar(PR)`
- `bar(PM)`
- `bar(QM)`
In the triangle PQR, `bar"PQ" = 2 bar" a" and bar"QR" = 2 bar"b"`. 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"`
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
