Given vectors, compute the step by step process of the given.

Question

Given vectors vecU=(-2,5,-4), vecV=(1,-6,3) and vecW=(2,3,1), compute the given:

1. ‖2vecU+3vecV-4vecW‖

Answer 1

Let u = <-2, 5, -4>, v = <1, -6, 3>, and w = <2, 3, 1>. We wish to compute ||2u + 3v - 4w||.

Compute each part of the sum independently using the scalar multiplication property of a vector.

2u = 2 <-2, 5, -4> = <-4, 10, -8>

3v = 3 <1, -6, 3> = <3, -18, 9>

4w = 4 <2, 3, 1> = <8, 12, 4>

Compute the sum directly using the addition property of vectors.

2u + 3v - 4w = <-4, 10, -8> + <3, -18, 9> - <8, 12, 4>

2u + 3v - 4w = <-4 + 3 - 8, 10 - 18 - 12, -8 + 9 - 4>

2u + 3v - 4w = <-9, -20, -3>

Compute the magnitude as the square root of the sum of the components squared.

||2u + 3v - 4w|| = √((-9)² + (-20)² + (-3)²)

||2u + 3v - 4w|| = √(81 + 400 + 9)

||2u + 3v - 4w|| = √(490)

||2u + 3v - 4w|| = 7√10