Question

By evaluating their dot product, find the values of the scalar s for which the two vectors ~b = ˆx + syˆ and ~c = ˆx−syˆ are orthogonal. (Remember that two vectors are orthogonal if and only if their dot product is zero.) Explain your answers with a sketch.

Answer 1

Answer:

$s=\pm 1$

Explanation:

The dot product of two vectors $\vec{a}=a_1\vec{x}+b_1\vec{y}$ and $\vec{b}=a_2\vec{x}+b_2\vec{y}$ is given by

$\vec{a}\cdot\vec{b}=a_1\cdot a_2+b_1\cdot b_2$

The dot product of two orthogonal vector is always zero thus if $\vec{a}=\vec{x}+s\vec{y}$ and $\vec{b}=\vec{x}-s\vec{y}$, their dot product would be

$\vec{a}\cdot\vec{b}=1\times1+s\times-(s)=1-s^2=0$

$\implies 1-s^2=0$

$\implies s^2=1$

$\implies s=\pm 1$