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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 another 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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