Question

Find two vectors in opposite directions that are orthogonal to the vector u. (There are many correct answers.)

Answer 1

Given:

The given vector is u=<7,9>.

Required:

We need to find two vectors in opposite directions that are orthogonal to the vector u.

Step-by-step explanation:

Recall that two vectors are orthogonal if their dot product equals 0.

Let v be the orthogonal vector to u.

`u\cdot v=0`
`<7,9>\cdot=0`
`7v_1+9v_2=0`
`7v_1+9v_2-9v_2=-9v_2`
`7v_1=-9v_2`
`(7v_1)/(7)=(-9v_2)/(7)`
`v_1=-(9)/(7)v_2`
`v=<-(9)/(7)v_2,v_2>`
`Let\text{ }v_2=7\text{ and substitute in the vector v.}`
`v=<-(9)/(7)*7,7>=<-9,7>`
`Let\text{ }v_2=-7\text{ and substitute in the vector v.}`
`v=<-(9)/(7)*(-7),-7>=<9,-7>`

Final answer:

`negative\text{ x-component, positive y-component=<-9,7>}`
`positive\text{ x-component, negative y-component=<9,-7>}`