Question

Let the vectors be u=(-1,0,2) , v=(0,2,-3) , w=(2,2,3) Calculate the following expressions a)<u,w> b) <2u- 5v,3w>

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Let's calculate the given expressions involving the vectors u, v, and w:
a) u • v (dot product of u and v):
u • v = (-1 * 0) + (0 * 2) + (2 * (-3)) = 0 - 0 - 6 = -6
b) <2u - 5v, 3w> (scalar triple product):
<2u - 5v, 3w> = (2u - 5v) • 3w
Now, calculate the individual components of 2u - 5v:
2u - 5v = (2 * (-1), 2 * 0, 2 * 2) - (5 * 0, 5 * 2, 5 * (-3)) = (-2, 0, 4) - (0, 10, -15) = (-2, -10, 19)
Now, take the dot product of (-2, -10, 19) and 3w:
(-2, -10, 19) • 3w = (-2 * 3, -10 * 3, 19 * 3) • (3 * 2, 3 * 2, 3 * 3) = (-6, -30, 57) • (6, 6, 9)
Now, calculate the dot product:
(-6, -30, 57) • (6, 6, 9) = (-6 * 6) + (-30 * 6) + (57 * 9) = -36 - 180 + 513 = 297
So, the scalar triple product <2u - 5v, 3w> is 297.

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