Determine the dot product of two vectors

The dot product of two vectors is the scalar obtained by multiplying corresponding components and adding the products, such as $\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y$ (and $+u_zv_z$ in three dimensions). It also represents $|\mathbf{u}||\mathbf{v}|\cos\theta$ , allowing interpretation of alignment: a zero dot product indicates perpendicular vectors, while its sign distinguishes acute from obtuse angles; abstract inner products and higher-dimensional generalizations are outside this scope.

Detailed Explanation: Determine the dot product of two vectors

To find the dot product, multiply corresponding components and add the results.

For two-dimensional vectors,

\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y.

For three-dimensional vectors, include the third pair:

\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z.

Example

Find the dot product of

\mathbf{u}=(3,-2) \qquad\text{and}\qquad \mathbf{v}=(4,5).

Step 1: Multiply corresponding components.

3(4)=12 \qquad\text{and}\qquad (-2)(5)=-10

Step 2: Add the products.

\mathbf{u}\cdot\mathbf{v}=12+(-10)=2

Therefore,

\boxed{\mathbf{u}\cdot\mathbf{v}=2}

The dot product is a scalar, not a vector. Its sign also gives information about the angle between the vectors:

A positive dot product means the angle is acute.
A dot product of $0$ means the vectors are perpendicular.
A negative dot product means the angle is obtuse.

Here, the dot product is positive, so the angle between $\mathbf{u}$ and $\mathbf{v}$ is acute.

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Vectors - Dot Product - Vector Notation to Dot Product

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