Question

Let `veca=<−2,3>` and `vecb =<−5,k>`. Find `k` so that `veca` and `vecb` will be orthogonal. Find k so that →a and →b will be orthogonal?

Let →a=⟨−2,3⟩ and →b =⟨−5,k⟩.

Find k so that →a and →b will be orthogonal.

Answer 1

`vec{a} \quad "and" \quad vec{b} \quad \ "will be orthogonal precisely when:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad k \ = \ -10/3.`

Explanation:

`"Recall that, for two vectors:" \qquad vec{a}, vec{b} \qquad "we have:"`

`\qquad vec{a} \quad "and" \quad vec{b} \qquad \quad "are orthogonal" \qquad \qquad hArr \qquad \qquad vec{a} cdot \vec{b} \ = \ 0.`

`"Thus:"`

`\qquad < -2, 3 > \quad "and" \quad < -5, k > \qquad \quad "are orthogonal" \qquad \qquad hArr`

`\qquad \qquad < -2, 3 > cdot < -5, k > \ = \ 0 \qquad \qquad hArr`

`\qquad \qquad \qquad ( -2 ) ( -5 ) + ( 3 ) ( k ) \ = \ 0 \qquad \qquad hArr`

`\qquad \qquad \qquad \qquad \qquad \qquad 10 + 3 k \ = \ 0 \qquad \qquad hArr`

`\qquad \qquad \qquad \qquad \qquad \qquad \quad 3 k \ = \ -10 \qquad \qquad hArr`

`\qquad \qquad \qquad \qquad \qquad \qquad \quad k \ = \ -10/3.`

`"So, from beginning to end here:"`

`\qquad < -2, 3 > \quad "and" \quad < -5, k > \qquad \quad "are orthogonal" \qquad \qquad hArr`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad k \ = \ -10/3.`

`"Thus, we conclude:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad k \ = \ -10/3.`