### Video Transcript

Given that π, π, and π are the direction cosines of a straight line, find the value of π squared plus π squared plus π squared.

Letβs just consider that we have the space π₯, π¦, and π§. Point π΄ is found at lowercase π₯π¦π§. And weβll have a line from the origin to point π΄, like this. Line ππ΄ is made up of three different vectors, here π₯ in pink, π¦ in yellow, and π§ in green.

Under these conditions, we can let π be equal to the cos of πΌ, where πΌ is the angle measure between this pink line and our vector ππ΄. π is gonna be equal to cos of π½, where π½ is the angle measure between the yellow line and blue line here. ~~π~~ [π] will be equal to the cosine of πΎ, where πΎ is the angle between the green line and the blue line in our figure.

And the cosine value will be equal to the adjacent side length π₯ over the length of our line, which we find by saying π₯ squared plus π¦ squared plus π§ squared and then taking the square root of that value. To find the cos of ~~B~~ [π½], weβll follow the same procedure. But this time, weβll have the length π¦ as the numerator. And the cos of πΎ will be equal to π§ over the square root of π₯ squared plus π¦ squared plus π§ squared.

Remember, weβre looking for π squared plus π squared plus π squared. If we plug in the values that we know for π, π, and π, we need to take π₯ squared. And then, we need to square the square root of π₯ squared plus π¦ squared plus π§ squared. And that will give us π₯ squared over π₯ squared plus π¦ squared plus π§ squared.

When we square π, we get π¦ squared over π₯ squared plus π¦ squared plus π§ squared. And when we square π, we get π§ squared over π₯ squared plus π¦ squared plus π§ squared. All three of these fractions have a common denominator, which means we can add their numerators. π₯ squared plus π¦ squared plus π§ squared canβt be simplified. So we just leave it π₯ squared plus π¦ squared plus π§ squared. And the denominator doesnβt change. Itβs also π₯ squared plus π¦ squared plus π§ squared. And when the numerator equals the denominator, the value of that fraction is one.

Weβve just shown that π squared plus π squared plus π squared equals one. In fact, the squares of the direction cosines of a straight line when added together always equal one.