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Dot Product of Vectors

The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

Projecting A Vector Onto A Vector

If the two vectors \( \vec{a} \) and \( \vec{b} \) are given by \( \vec{a} = 4i + 3j \) and \( \vec{b} = -3i + j, \) what is the difference between the length of the projection of \( \vec{b} \) onto \( \vec{a} \) and the length of the projection of \( \vec{a} \) onto \( \vec{b}? \)

A line \(l\) passes through the point \( P(0, 4).\) Let \( A\) and \(B\) be the two intersection points of \(l\) and the circle \( x^2 + y^2= 64 .\) If \(O\) is the origin and \( Q \) is the intersection point of the two tangent lines at points \( A\) and \( B, \) what is the result of the dot product \( \overrightarrow{OP} \cdot \overrightarrow{OQ}?\)

Project \(\vec{ u} \) onto \( \vec{v}, \) both defined below, and write the resulting vector in the same form as \( \vec{v}:\) \[ \vec{u} = i - 2j \text{ and } \vec{v} = -10 i. \]

Calculate the vector projection of \( \vec{a} = (-2, 3) \) onto \( \vec{b} = (6, 8). \)

If the two vectors \( \vec{a} \) and \( \vec{b} \) are given by \( \vec{a} = 3i + 4j \) and \( \vec{b} = -2i + 4j, \) what is the difference between the length of the projection of \( \vec{b} \) onto \( \vec{a} \) and the length of the projection of \( \vec{a} \) onto \( \vec{b}? \)

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