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.

1 Answer
Mar 2, 2018

Answer:

# 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. #