Question

Find a unit vector that is orthogonal to both i + j and i + k

Answer 1

Vectors:

i = [1 0 0]

j = [0 1 0]

k = [0 0 1]

i + j = [1 1 0]

i + k = [1 0 1]

We want a vector which is orthogonal to both i + j and i + k, so the escalar procut between our vector "v" and i + j, i + k is equal to 0

Let's say v = [x y z]

v . (i + j) = [x y z] . [1 1 0] = x.1 + y.1 + z.0 = 0 => x + y = 0

v . (i + k) = [x y z] . [1 0 1] = x.1 + y.0 + z.1 = 0 => x + z = 0

So, we can say:

x + y = 0 => x = -y

x + z = 0 => x = -z

If x = -y:

x + z = 0

-y + z = 0

z = y = a and x = -a

We know it's a unit vector, so:

$\sqrt{x^2+y^2+z^2}=1$

$\sqrt{(-a)^2+a^2+a^2}=1$

$\sqrt{a^2+a^2+a^2}=1$

$\sqrt{3a^2}=1$

$|a|\sqrt{3}=1$

$|a|=\frac{1}{\sqrt{3}}$

So,

y = z = a = $\frac{1}{\sqrt{3}}$ and x = -a = $-\frac{1}{\sqrt{3}}$

v = [ $-\frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}} \frac{1}{\sqrt{3}}$ ]