# Question Video: Finding the Projection of a Vector in the Direction of Another given Their Norms and the Angle between Them Mathematics • 12th Grade

If |π| = 5, |π| = 15, and the measure of the angle between them is 30Β°, find the algebraic projection of π in the direction of π.

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### Video Transcript

If the magnitude of π equals five, the magnitude of π equals 15, and the measure of the angle between them is 30 degrees, find the algebraic projection of π in the direction of π.

Alright, so in this exercise, we have these two vectors π and π. And we know that the magnitude of π is one-third the magnitude of π and also that theyβre separated by an angle of 30 degrees. Knowing this, we want to solve for the algebraic projection, also known as the scalar projection, of vector π in the direction of vector π. What weβre proposing then is effectively to lay vector π down along this line weβve drawn through vector π. Essentially, weβre solving for how much of vector π is parallel to vector π.

We can figure out how to compute this scalar projection in one of two ways. The first way is to recall from memory that the scalar projection of one vector, weβve called it π one, onto another, π two, is given by these two expressions here. So, for example, we could use the fact that the magnitude of the first vector multiplied by the cosine of the angle between the two vectors gives us this scalar projection to solve for the algebraic projection of π onto π.

Notice though that we could also reach this conclusion based on our sketch of vectors π and π. The vector π essentially forms the hypotenuse of a right triangle. And so we see that itβs the magnitude of that hypotenuse, the magnitude of π, multiplied by the cosine of the angle between π and the direction of π, which is 30 degrees, that gives us this algebraic projection. Either method we choose leads us to the same result.

We substitute in the magnitude of π, 15. And knowing that the cos of 30 degrees equals exactly the square root of three over two, we find our answer to be 15 times the square root of three over two. This is the algebraic or scalar projection of π in the direction of π.