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.
