The Scalar Product (DP IB Analysis & Approaches (AA): HL): Revision Note

The scalar (dot) product

What is the scalar product?

• The scalar product is an operation which takes two vectors and outputs a scalar

• The scalar product between two vectors $\mathit{v}$ and $\mathit{w}$ is denoted $\mathit{v}·\mathit{w}$

  • This is why it is also called the dot product

How is the scalar product calculated?

• One formula for the scalar product is $\mathit{v}·\mathit{w}= v_1{w}_1+ v_2{w}_2+ v_3{w}_3$

  • $\mathit{v}= \left(\genfrac{}{}{0pt}{}{v_1}{\begin{array}{c}{v}_2\\ v_3\end{array}}\right)$

  • $\mathit{w}= \left(\genfrac{}{}{0pt}{}{w_1}{\begin{array}{c}{w}_2\\ w_3\end{array}}\right)$

• Another formula for the scalar product is $\mathit{v}·\mathit{w}=\left|\mathit{v}\right|\left|\mathit{w}\right|\cos \theta$

  • $\theta$ is the angle between $\mathit{v}$ and $\mathit{w}$

What properties of the scalar product do I need to know?

• The scalar product is commutative

  • $\mathit{v}·\mathit{w}=\mathit{ }\mathit{w}·\mathit{v}$

• The distributive law over addition can be used to expand brackets

  •  $\mathit{u}\cdot \left(\mathit{v}+\mathit{w}\right)=\mathit{u}·\mathit{v}+\mathit{u}·\mathit{w}$

• The scalar product is associative with respect to multiplication by a scalar

  • $\left(k\mathit{v}\right)\cdot \left(\mathit{w}\right)=\mathit{v}·\left(k\mathit{w}\right)=k(\mathit{v}\cdot \mathit{w})$

• The scalar product between a vector and itself is equal to the square of its magnitude

  • $\mathit{v}·\mathit{v}={\left|\mathit{v}\right|}^2$

• The absolute value of the scalar product of two parallel vectors is equal to the product of their magnitudes

  • $|\mathit{v}·\mathit{w}|=\left|\mathit{w}\right|\left|\mathit{v}\right|$

    • This is because $\cos 0°=1$ and $\cos 180°=-1$

• The scalar product of two perpendicular vectors is equal to zero

  • This is because $\cos 90°=0$