Question Video: Determining the Angle between Two Vectors

Video Transcript

Find the angle between the two vectors 𝐲 equals 2.0𝐢 plus 4.0𝐣 plus 8.0𝐤 and 𝐳 equals 6.0𝐢 plus 4.0𝐣 plus 6.0𝐤.

If we call the angle between these two vectors 𝜃, it’s that we want to solve for. To start out, we can recall that the dot product between two vectors, 𝐀 and 𝐁, is equal to the product of their magnitudes times the cosine of the angle between them. And we can recall further that the dot product is also equal to the product of the 𝑥-components plus the product of the 𝑦-components plus the product of the 𝑧-components of our two vectors. Bringing those two relationships together, we can write that, in our case, the product of the 𝑥-components of our vectors plus the product of the 𝑦-components plus the product of the 𝑧-components. Is equal to the product of their magnitudes multiplied by the cos of 𝜃, the angle between them. If we divide both sides of this equation by the magnitude of 𝐲 times the magnitude of 𝐳 and then take the inverse cosine of both sides. So 𝜃, the angle we’re interested in, is equal to the inverse cos of 𝑦 𝑥, 𝑧 𝑥 plus 𝑦 𝑦, 𝑧 𝑦 plus 𝑦 𝑧, 𝑧 𝑧 divided by the product of the magnitudes of our two vectors.

We can recall that the magnitude of a vector, when it has three dimensions, is equal to the square root of the 𝑥-dimension squared plus the 𝑦-dimension squared plus the 𝑧-dimension squared. If we insert the magnitude expansion for both 𝑦 and 𝑧, then we now have an expression for 𝜃 entirely in terms of the components of our two vectors. In the case of our vector 𝐲, 2.0 is 𝑦 𝑥, 4.0 is 𝑦 𝑦, and 8.0 is 𝑦 𝑧. And for 𝐳, 6.0 is 𝑧 𝑥, 4.0 is 𝑧 𝑦, and 6.0 is 𝑧 𝑧. When we plug each of these values in where it fits in our equation. With all these values plugged in, when we enter this expression on our calculator, we find 𝜃 is 28 degrees. That’s the angle between our two vectors 𝐲 and 𝐳.