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Find the scalar and vector projections of $ b $ onto $ a $.

$ a = \langle -5, 12 \rangle $ , $ b = \langle 4, 6 \rangle $

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01:50

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

Missouri State University

Baylor University

University of Nottingham

Idaho State University

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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So for us to find these two projections, we just need to plug everything into those formulas they give us in the chapter. So notice how both of them have the dot product of a with B as well as, um, the magnitude of a. So we'll need to go out and find both of those. I'll find the magnitude of a first because that's a little bit easier to do. So we're going to square each of the components Adam together. So negative. Five square plus, um, about 12 12 squared, all square rooted. So that would be 25 plus 1 44 which is 1 69. Just 13. Now to get the dot product So a dotted with B, remember, we're going to multiply each of these components together and then add the results, so we're gonna do negative five times for, and then plus 12 times six. So that would be negative. 20 plus 72 which is 52. Now we just need to plug these in, so over here, it's going to be 50 to over 13, which simplifies down to four. Um, so this is our scalar projection. And then for the factor projection. So again we have 15 or 52 over, uh, will be 13 squares that I just be 13 times 13. And then we have our victor A which is negative 5. 12. So that should simplify down to 4/13, and then we just go ahead and distribute. So that would be negative. 33rd and then 48 13th and then this Here is our sector projection.

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