We know that two vectors are ortogonal if and only if:
[tex]\vec{v}\cdot\vec{w}=0[/tex]
where
[tex]\vec{v}\cdot\vec{w}=v_1w_1+v_2w_2[/tex]
is the dot product between the vectors.
In this case we have the vectors:
[tex]\begin{gathered} \vec{a}=\langle-4,-3\rangle \\ \vec{b}=\langle-1,k\rangle \end{gathered}[/tex]
the dot product between them is:
[tex]\begin{gathered} \vec{a}\cdot\vec{b}=(-4)(-1)+(-3)(k) \\ =4-3k \end{gathered}[/tex]
and we want them to be ortogonal, so we equate the dot product to zero and solve the equation for k:
[tex]\begin{gathered} 4-3k=0 \\ 4=3k \\ k=\frac{4}{3} \end{gathered}[/tex]
Therefore, for the two vector to be ortogonal k has to be 4/3.
