Question

part f
consider the vectors \\( \mathbf{u} = \langle -1, 0 \
angle \\), \\( \mathbf{v} = \langle 0, 3 \
angle \\), and \\( \mathbf{w} = \langle 4, 4 \
angle \\). find these resultant vectors:
\\( 2\mathbf{u} - 2\mathbf{v} + \mathbf{w} \\)
\\( -\mathbf{u} + 3\mathbf{v} + 2\mathbf{w} \\)

Explanation:

Step1: Calculate \(2\mathbf{u} - 2\mathbf{v} + \mathbf{w}\)

First, find \(2\mathbf{u}\), \(-2\mathbf{v}\) by scalar multiplication. For a vector \(\mathbf{a}=\langle a_1,a_2
angle\) and scalar \(k\), \(k\mathbf{a}=\langle ka_1,ka_2
angle\).
\(2\mathbf{u}=2\langle - 1,0
angle=\langle - 2,0
angle\)
\(-2\mathbf{v}=-2\langle0,3
angle=\langle0, - 6
angle\)
\(\mathbf{w}=\langle4,4
angle\)
Now, add the vectors component - wise: \(\langle - 2,0
angle+\langle0, - 6
angle+\langle4,4
angle=\langle-2 + 0+4,0-6 + 4
angle=\langle2,-2
angle\)

Step2: Calculate \(-\mathbf{u}+3\mathbf{v}+2\mathbf{w}\)

First, find \(-\mathbf{u}\), \(3\mathbf{v}\), \(2\mathbf{w}\) by scalar multiplication.
\(-\mathbf{u}=-\langle - 1,0
angle=\langle1,0
angle\)
\(3\mathbf{v}=3\langle0,3
angle=\langle0,9
angle\)
\(2\mathbf{w}=2\langle4,4
angle=\langle8,8
angle\)
Now, add the vectors component - wise: \(\langle1,0
angle+\langle0,9
angle+\langle8,8
angle=\langle1 + 0+8,0 + 9+8
angle=\langle9,17
angle\)

Answer:

For \(2\mathbf{u}-2\mathbf{v}+\mathbf{w}\), the resultant vector is \(\langle2,-2
angle\); for \(-\mathbf{u}+3\mathbf{v}+2\mathbf{w}\), the resultant vector is \(\langle9,17
angle\)