Home
Class 12
MATHS
Find a unit vector perpendicular to t...

Find a unit vector perpendicular to the plane determined by the points `(1,-1,2),(2,0,-1)a n d(0,2,1)dot`

Text Solution

Verified by Experts

The correct Answer is:
`+- .((2hat(i)+hat(j)+hat(k)))/(sqrt(6))`
A unit vector perpendicular to plane determine by
`P,Q,R =+- ((vec(PQ))xx(vec(PR)))/(|vec(PQ) xx vec(PR)|)`
`:. " Unit vector " =+- ((vec(PQ)) xx (vec(PR)))/(|vec(PQ) xx vec(PR)|)`
where `vec(PQ) = hat(i) + hat(j) - 3hat(k)`
and `vec(PR) = - hat(i) + 3hat(j) - hat(k)`
`:. vec(PQ) xx vec(PR) = |{:(hat(i),,hat(j),,hat(k)),(1,,1,,-3),(-1,,3,,-1):}|`
`=hat(i) (-1 +9) -hat(j) (-1-3) + hat(k) (3+1)`
`=8i + 4j+ 4hat(k)`
`rArr |vec(PQ)xx vec(PR)| = sqrt(4+1+1) =4 sqrt(6)`
`:. (vec(PQ)xxvec(PR))/(|vec(PQ)xxvec(PR)|) =+- (4(2hat(i)+hat(j)+hat(k)))/(4sqrt(6))`
`=+- (2hat(i) +hat(j) + hat(k))/(sqrt(6))`
Promotional Banner

Similar Questions

Explore conceptually related problems

The unit vector perendicular to the plane determined by P (1,-1,2) ,C(3,-1,2) is ________.

Find the area of the square formed by the points (0, -1), (2, 1) (0, 3) and (-2, 1) as vertices.

Find the equation of the line passing through the points (1,2,3)a n d(-1,0,4)dot

Find the vector and cartesian equation of the line joining the points (1, -2, 1) and (0, -2, 3).

Find the vector and cartesian equations of the plane passing through the points A(1, -2, 3) and B(-1,2,-1) and is parallel to the line (x-2)/(2)=(y+1)/(3)=(z-1)/(4) .

Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D(0, 0, 2) . Point D lies on a line L orthogonal to the plane determined by the points A, B and C.

Find the equation of the plane passing through the points (3, 4, 2), (2, -2, -1) and (7, 0, 1).

Find the vector and certesian equation of the plane through the points (1, 2, 3) and (2, 3, 1) and perpendicular to the plane vecr*(3hati-2hatj+4hatk)=5 .

Find the equation of the plane passing through the points (1,0,-1)a n d(3,2,2) and parallel to the line x-1=(1-y)/2=(z-2)/3dot

Find the vector and Cartesian equations of the plane passing through the point (1,1,-1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0