Question

given vector \\(\mathbf{u} = \langle -1, -1 \
angle\\) and \\(\mathbf{v} = \langle -3, 1 \
angle\\), determine the value of \\(4\mathbf{u} - 4\mathbf{v}\\) in component form.

Explanation:

Step1: Factor out the scalar multiple

We can factor out the 4 from the expression \(4\mathbf{u}-4\mathbf{v}\) using the distributive property of scalar multiplication over vector subtraction. The distributive property states that \(a\mathbf{u}-a\mathbf{v}=a(\mathbf{u}-\mathbf{v})\) for a scalar \(a\) and vectors \(\mathbf{u}\) and \(\mathbf{v}\). So, \(4\mathbf{u}-4\mathbf{v} = 4(\mathbf{u}-\mathbf{v})\).

Step2: Subtract the vectors \(\mathbf{u}\) and \(\mathbf{v}\)

To subtract two vectors \(\mathbf{u}=\langle u_1,u_2
angle\) and \(\mathbf{v}=\langle v_1,v_2
angle\), we use the formula \(\mathbf{u}-\mathbf{v}=\langle u_1 - v_1,u_2 - v_2
angle\). Given \(\mathbf{u}=\langle - 1,-1
angle\) and \(\mathbf{v}=\langle - 3,1
angle\), we have:
\(\mathbf{u}-\mathbf{v}=\langle-1-(-3),-1 - 1
angle=\langle-1 + 3,-2
angle=\langle2,-2
angle\)

Step3: Multiply the resulting vector by the scalar 4

To multiply a vector \(\langle x,y
angle\) by a scalar \(a\), we use the formula \(a\langle x,y
angle=\langle ax,ay
angle\). Here, \(a = 4\) and the vector is \(\langle2,-2
angle\), so:
\(4\langle2,-2
angle=\langle4\times2,4\times(-2)
angle=\langle8,-8
angle\)

Answer:

\(\langle 8, - 8
angle\)