Let U2i 3j4k Vi 3j k compute UxV and also show that the vec

Let U=2i +3j+4k, V=-i +3j - k, compute U_xV, and also show that the vector U_xV is orthogonal to both the vectors U and V. Find an equation of a plane passing through three points (2. 3,4),(-1,-2,3).(-5,-4,2)

Solution

10 . cross product solution is correct.

Let cross product W = (-15 , -2, 9)

To prove both are orthogonal take dot procut of both vectors:

U = (2 , 3 , 4) ; V = .( -1 ,3 , -1)

U.W = ( -30 -6 +36 =0)

V.W = (15 -6 -9 =0

As the dot product of UxV and U and V is zero, so they are orthogonal.

11.) U = P1P2 = -3i -5j -k

V = -7i -7j -2k

U x V = 3i + j -14k

equation of plane : ax +by +cz +d =0

3x+y-14z +d =0

sustitute any one point to find d :

3*2 + 3 -14*4 +d=0

d +6 +3 - 56 =0

d =47

So, equation of plane :; 3x+y-14z + 47=0