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

Question

Let →a=⟨−2,3⟩ and →b =⟨−5,k⟩.

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

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Answer 1 · 2018-03-02T00:19:41

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

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

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

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

$"Thus:"$

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

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

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

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

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

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

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

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

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

$"Thus, we conclude:"$

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