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.

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$$\frac{7v_1}{7}=\frac{-9v_2}{7}$$v_1=-\frac{9}{7}v_2$$v=<-\frac{9}{7}v_2,v_2>$$Let\text{ }v_2=7\text{ and substitute in the vector v.}$$v=<-\frac{9}{7}\times7,7>=<-9,7>$$Let\text{ }v_2=-7\text{ and substitute in the vector v.}$$v=<-\frac{9}{7}\times(-7),-7>=<9,-7>$

Final answer:

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