Let 𝐚 = (10, -6, 4) and 𝐛 = (7, 7, 3) be vectors. (A) Find...

Let \(\mathbf{a} = (10, -6, 4)\) and \(\mathbf{b} = (7, 7, 3)\) be vectors. (A) Find the scalar projection of \(\mathbf{b}\) onto \(\mathbf{a}\). Scalar Projection: (B) Decompose the vector \(\mathbf{b}\) into a component parallel to \(\mathbf{a}\) and a component orthogonal to \(\mathbf{a}\). Parallel component: Orthogonal Component:

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00:01 Okay, we are going to find the vector projection of v into w.

00:06 So let's just talk about some of our formulas here before we start.

00:12 Now, we can look at our dot product formula.

00:16 It will help us consider kind of different forms of our projection of vnw.

00:23 So a normal projection of v onto w would just be the magnitude of v times the cosine of theta.

00:32 Well, using the formula above, you can divide both sides by the magnitude of w.

00:37 So you can see that that formula can also be the dot product divided by the magnitude of w.

00:43 So that is the projection of vnw, but notice they want us to find a vector projection of vnw.

00:51 So we are going to have to multiply by our vector projection.

01:01 And we'll change that formula here in a second.

01:04 Let's go ahead and find the dot product piece.

01:07 Now notice we didn't have a k piece written in our vector v, and we didn't have a i piece written in our vector w.

01:17 So we did add those so that we can bring those in.

01:22 But look it makes two things zero out, and we really just have a dot product of a negative 1.

01:38 Okay, so we will have negative 1 over and then we need our magnitude of w.

01:44 We'll notice we have a 1 and a 1...

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