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

`s=\pm 1`

Step-by-step 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*1+s*-(s)=1-s^2=0`

`\implies 1-s^2=0`

`\implies s^2=1`

`\implies s=\pm 1`