Question

given the vectors ( mathbf{a}=langle-6,7
angle ) and ( mathbf{b}=langle-3,0
angle ), find ( 6 mathbf{a} ) and ( mathbf{a}+mathbf{b} ).
write your answers in component form.
( 6 mathbf{a}=langlesquare,square
angle )
( mathbf{a}+mathbf{b}=langlesquare,square
angle )

Explanation:

Step1: Calculate \(6\mathbf{a}\)

Given \(\mathbf{a}=\langle - 6,7
angle\), to find \(6\mathbf{a}\), we use the scalar - vector multiplication formula \(k\langle x,y
angle=\langle kx,ky
angle\).
Here \(k = 6\), \(x=-6\), \(y = 7\).
So \(6\mathbf{a}=6\langle - 6,7
angle=\langle6\times(-6),6\times7
angle\)

$$6\mathbf{a}=\langle - 36,42 angle$$

Step2: Calculate \(\mathbf{a}+\mathbf{b}\)

Given \(\mathbf{a}=\langle - 6,7
angle\) and \(\mathbf{b}=\langle - 3,0
angle\), we use the vector - addition formula \(\langle x_1,y_1
angle+\langle x_2,y_2
angle=\langle x_1 + x_2,y_1 + y_2
angle\)
Here \(x_1=-6\), \(y_1 = 7\), \(x_2=-3\), \(y_2 = 0\)
\(\mathbf{a}+\mathbf{b}=\langle-6+( - 3),7 + 0
angle\)

$$=\langle-6-3,7 angle=\langle - 9,7 angle$$

Answer:

\(6\mathbf{a}=\langle - 36,42
angle\)
\(\mathbf{a}+\mathbf{b}=\langle - 9,7
angle\)