Question

Find a vector of magnitude 15, which is perpendicular to both the vectors `(4hat(i) -hat(j)+8hat(k)) and (-hat(j)+hat(k)).`

Answer 1

Let `vec(a) = (4 hat(i)-hat(j) + 8 hat(k) ) and vec(b)= (-hat(j) + hat(k)).`
A unit vector perpendicular to both `vec(a) and vec(b) = ((vec(a) xxvec(b)))/(|vec(a) xx vec(b)|`.
Now, `vec(a) xx vec(b) = |{:(hat(i),hat(j),hat(k)),(4,-1,8),(0,-1,1):}|`
`=(-1+8) hat(i) - ( 4-0) hat(j) + (-4-0) hat(k)`
`=(7 hat(i) - 4 hat(j) - 4 hat(k)).`
`:. |vec(a) xxvec(b)| = sqrt(7^(2) + (-4)^(2)+(-4)^(2)) =sqrt(81)=9.`
So, a unit vector perpendicular to both `vec(a) and vec(b)`.
`((vec(a) xxvec(b)))/|(vec(a) xx vec(b))| = (7 hat(i) -4 hat(j)- 4 hat(k))/9.`
The required vector `=(15(7hat(i)-4hat(j)-4hat(k)))/9 =5/3(7 hat(i) - 4 hat(j) - 4 hat(k)).`