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If $ \theta $ is the angle between vectors $ a $ and $ b $, show that$$ proj_a b \cdot proj_b a = (a \cdot b) \cos^2 \theta $$

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Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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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The problem is: if theta is angle between vectors, a and b showed that to work the projection of b into a net projection of work to a unto b is equal to a delta b times. Cosine theta square first by definition, vector projection of fluoro. A is equal to a dot over the magnitude of a square times. A and its vector projection of a on to b is equal to b dot a over magnitude of v square times p. So this 1 out of this 1. But this is equal to a t, has e dot a over addingtude of a square times magnitude of v square times a dot b. So this is equal to magnitude of a times magnitude of b times cosine theta square over magnitude of a square times, magnitude b square times a dot b. So this is equal to a dot b times, cosine theta score.

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