find the triple scalar product u . (v * w) for the vectors
u= (5, 9, -6)
v= (5, 8, -1)
w= (-2, 8, 3)
if by v * w , you mean
v X w , the cross product, then
v X w = (5, 8, -1) X (-2, 8, 3)
= (32, -13, 56) , I assume you know how to find the cross-product
Then u . (v * w)
= (5, 9, -6) . (32, -13, 56)
= 160 - 117 - 336
= - 293
u • v × w is just the determinant
|5 9 -6|
|5 8 -1|
|-2 8 3|
To find the triple scalar product u . (v * w) for the given vectors u, v, and w, you need to follow these steps:
Step 1: Find the cross product (v * w)
To find the cross product, you can use the formula:
(v * w) = (v2w3 - v3w2, v3w1 - v1w3, v1w2 - v2w1)
So, for the given vectors v and w, the cross product (v * w) will be:
(v * w) = ((8 * 3) - (-1 * 8), (-1 * -2) - (5 * 3), (5 * 8) - (8 * -2))
Step 2: Calculate the dot product u . (v * w)
To calculate the dot product, you can use the formula:
u . (v * w) = u1(v * w)1 + u2(v * w)2 + u3(v * w)3
So, for the given vectors u and (v * w), the dot product u . (v * w) will be:
u . (v * w) = (5 * ((8 * 3) - (-1 * 8))) + (9 * ((-1 * -2) - (5 * 3))) + (-6 * ((5 * 8) - (8 * -2)))
Simplifying the expression:
u . (v * w) = (5 * (24 + 8)) + (9 * (2 - 15)) + (-6 * (40 + 16))
Further simplifying:
u . (v * w) = (5 * 32) + (9 * -13) + (-6 * 56)
u . (v * w) = 160 - 117 - 336
u . (v * w) = -293
Therefore, the triple scalar product u . (v * w) for the given vectors u, v, and w is -293.