Advertisements
Advertisements
Question
If `vec"a", vec"b", vec"c"` determine the vertices of a triangle, show that `1/2[vec"b" xx vec"c" + vec"c" xx vec"a" + vec"a" xx vec"b"]` gives the vector area of the triangle. Hence deduce the condition that the three points `vec"a", vec"b", vec"c"` are collinear. Also find the unit vector normal to the plane of the triangle.
Solution
Since, `vec"a", vec"b"` and `vec"c"` are the vertices of ΔABC
∴ `vec"AB" = vec"b" - vec"a"`
`vec"BC" = vec"c" - vec"b"`
And `vec"AC" = vec"c" - vec"a"`
∴ Area of ΔABC = `1/2 |vec"AB" xx vec"AC"|`
= `1/2|(vec"b" - vec"a") xx (vec"c" - vec"a")|`
= `1/2 |vec"b" xx vec"c" - vec"b" xx vec"a" - vec"a" xx vec"c" + vec"a" xx vec"a"|`
= `1/2 |vec"b" xx vec"c" + vec"a" xx vec"b" + vec"c" xx vec"a"|` ......`[(because vec"a" xx vec"b" = - vec"b" xx vec"a"),(vec"c" xx vec"a" = - vec"a" xx vec"c"),(vec"a" xx vec"a" = vec0)]`
For three vectors are collinear, area of ΔABC = 0
∴ `1/2 |vec"b" xx vec"c" + vec"a" xx vec"b" + vec"c" xx vec"a"|` = 0
`|vec"a" xx vec"b" + vec"b" xx vec"c" + vec"c" xx vec"a"|` = 0
Which is the condition of collinearity of `vec"a", vec"b"` and `vec"c"`.
Let `hat"n"` be the unit vector normal to the plane of the ΔABC
∴ `hat"n" = (vec"AB" xx vec"AC")/|vec"AB" xx vec"AC"|`
⇒ `(vec"a" xx vec"b" + vec"b" xx vec"c" + vec"c" xx vec"a")/|vec"a" xx vec"b" + vec"b" xx vec"c" + vec"c" xx vec"a"|`
APPEARS IN
RELATED QUESTIONS
If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors such that \[x \vec{a} + y \vec{b} = \vec{0} ,\] then write the values of x and y.
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 \[\overrightarrow{a}\] is a non-zero vector of modulus a and m is a non-zero scalar such that m \[\overrightarrow{a}\] is a unit vector, write the value of m.
Write a unit vector in the direction of \[\overrightarrow{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .\]
Find the position vector of the mid-point of the line segment AB, where A is the point (3, 4, −2) and B is the point (1, 2, 4).
Write a unit vector in the direction of \[\overrightarrow{PQ}\], where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.
The vector equation of the plane passing through \[\vec{a} , \vec{b} , \vec{c} ,\text{ is }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} ,\] provided that
If three points A, B and C have position vectors \[\hat{i} + x \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k}\text{ and }y \hat{i} - 2 \hat{j} - 5 \hat{k}\] respectively are collinear, then (x, y) =
If \[\vec{a}\text{ and }\vec{b}\] are two collinear vectors, then which of the following are incorrect?
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 area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).
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:
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 two sides of a triangle are `hat"i" + 2hat"j" and hat"i" + hat"k"`, find the length of the third side.
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 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`.
Find the acute angle between the curves at their points of intersection, y = x2, y = x3.
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" 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")`
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.
The XZ plane divides the line segment joining the points (3, 2, b) and (a, -4, 3) in the ratio ______.
lf `overlinea` and `overlineb` be two unit vectors and θ is the angle between them, then `|overlinea - overlineb|` is equal to ______
Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if ______.
If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.
If `veca` and `vecb` are two collinear vectors then which of the following are incorrect.
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"`.
In the triangle PQR, `bar(PQ) = 2bara and bar(QR) = 2barb`. The mid-point of PR is M. Find the following vectors in terms of `bara and barb`.
- `bar(PR)`
- `bar(PM)`
- `bar(QM)`
