Question

Find k so that a and b will be orthogonal

Answer 1

k = -5

Explanation:

The scalar product of vectors a and b is zero when the vectors are orthogonal (perpendicular):

`\boxed{\begin{aligned}\vec a \cdot \vec b & = |a||b| \cos \theta\\& = |a||b| \cos 90^(\circ)\\& = 0 \end{aligned}}`

`\textsf{Therefore, $\vec a$\; and \;$\vec b$\; are orthogonal when\; $\vec a\cdot \vec b=0$}`

Given:

• a = ⟨2, -2⟩
• b = ⟨−5, k⟩

Substitute the given vectors into the formula and solve for k:

`\implies \langle 2, -2 \rangle \cdot \langle-5, k \rangle=0`

`\implies (2)(-5) +(-2)(k)=0`

`\implies -10-2k=0`

`\implies 2k=-10`

`\implies k=-5`