Question

find two vectors orthogonal to the given vector.
u = \langle 4,2\
angle

Explanation:

Step1: Recall the dot - product formula

If two vectors \(\mathbf{v}=\langle x_1,y_1
angle\) and \(\mathbf{u}=\langle x_2,y_2
angle\) are orthogonal, then \(\mathbf{v}\cdot\mathbf{u}=x_1x_2 + y_1y_2=0\). Let \(\mathbf{v}=\langle x,y
angle\) and \(\mathbf{u}=\langle4,2
angle\). So we have \(4x + 2y=0\), which can be rewritten as \(y=-2x\).

Step2: Choose values for \(x\)

• Let \(x = 1\). Then \(y=-2(1)=-2\). So one vector \(\mathbf{v}_1=\langle1,-2

angle\).

• Let \(x=-1\). Then \(y=-2(-1) = 2\). So another vector \(\mathbf{v}_2=\langle-1,2

angle\).

Answer:

\(\langle1,-2
angle\) and \(\langle-1,2
angle\)