Let a=[−3,0,−3]𝑎=[−3,0,−3], b=[0,−1,0]𝑏=[0,−1,0], and c=[0,3,−2]𝑐=[0,3,−2]. Compute (a×b)⋅c(𝑎×𝑏)⋅𝑐.

Sure, let's break this down step by step.

First, we need to compute the cross product of vectors a and b, denoted as a×b. The cross product of two vectors in 3D space is given by:

a×b = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1]

where a1, a2, a3 and b1, b2, b3 are the components of vectors a and b respectively.

So, for a = [-3, 0, -3] and b = [0, -1, 0], we have:

a×b = [00 - (-3)(-1), -30 - (-3)0, -3(-1) - 00] = [0, 0, -3]

Next, we need to compute the dot product of the result (a×b) and vector c, denoted as (a×b)⋅c. The dot product of two vectors is given by:

(a×b)⋅c = a1c1 + a2c2 + a3c3

where a1, a2, a3 are the components of vector (a×b) and c1, c2, c3 are the components of vector c.

So, for (a×b) = [0, 0, -3] and c = [0, 3, -2], we have:

(a×b)⋅c = 00 + 03 + (-3)*(-2) = 6

So, the result of (a×b)⋅c is 6.