Product of Two Vectors

10.6 Product of Two Vectors

So far we have studied addition and subtraction of vectors. Multiplication of two vectors is defined in two ways — one where the result is a scalar (the scalar or dot product), and another where the result is a vector (the vector or cross product). Both have wide applications in geometry, mechanics, and engineering.

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Scalar (Dot) Product

The scalar product of two vectors $\vec{a}$ and $\vec{b}$ is defined as:

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, with $0 \le \theta \le \pi$.

If either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$, then $\theta$ is not defined, and we define $\vec{a} \cdot \vec{b} = 0$.

Observations

1. $\vec{a} \cdot \vec{b}$ is positive if $\theta$ is acute ($0 \le \theta < \frac{\pi}{2}$).
2. $\vec{a} \cdot \vec{b}$ is zero if $\theta = \frac{\pi}{2}$.
3. $\vec{a} \cdot \vec{b}$ is negative if $\theta$ is obtuse ($\frac{\pi}{2} < \theta \le \pi$).

Properties of the Dot Product

Property 1 (Commutativity): $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

›Proof

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = |\vec{b}| |\vec{a}| \cos \theta = \vec{b} \cdot \vec{a}$, since real-number multiplication is commutative.

Property 2 (Distributivity over addition): $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

›Proof

Using the dot product as $|\vec{a}|$ times the projection of the other vector onto $\vec{a}$: the projection of $\vec{b}+\vec{c}$ onto $\vec{a}$ equals the sum of the projections of $\vec{b}$ and $\vec{c}$ onto $\vec{a}$. Multiplying by $|\vec{a}|$ gives the result.

Property 3 (Scalar multiplication): $(\lambda \vec{a}) \cdot \vec{b} = \lambda (\vec{a} \cdot \vec{b}) = \vec{a} \cdot (\lambda \vec{b})$

›Proof

For $\lambda \ge 0$: $(\lambda \vec{a}) \cdot \vec{b} = |\lambda| |\vec{a}| |\vec{b}| \cos \theta = \lambda (\vec{a} \cdot \vec{b})$. For $\lambda < 0$, the angle between $\lambda \vec{a}$ and $\vec{b}$ is $\pi - \theta$, and $\cos(\pi - \theta) = -\cos \theta$, so $(\lambda \vec{a}) \cdot \vec{b} = |\lambda| |\vec{a}| |\vec{b}| (-\cos \theta) = \lambda (\vec{a} \cdot \vec{b})$. The same reasoning applies to $\vec{a} \cdot (\lambda \vec{b})$.

Property 4 (Dot product with itself): $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

›Proof

$\vec{a} \cdot \vec{a} = |\vec{a}| |\vec{a}| \cos 0 = |\vec{a}|^2$.

Dot Product in Component Form

For $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$ and $\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$, using $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ and $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$:

Angle Between Two Vectors

Projection of a Vector

The scalar projection of $\vec{a}$ onto $\vec{b}$ (the component of $\vec{a}$ along $\vec{b}$) is:

The vector projection of $\vec{a}$ onto $\vec{b}$ is:

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Vector (Cross) Product

The vector product of two vectors $\vec{a}$ and $\vec{b}$ is defined as:

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ ($0 \le \theta \le \pi$), and $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, such that $\vec{a}, \vec{b}, \hat{n}$ form a right-handed system.

If either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$, then $\theta$ is not defined, and we define $\vec{a} \times \vec{b} = \vec{0}$.

The direction of $\vec{a} \times \vec{b}$ is given by the right-hand rule: if you curl the fingers of your right hand from $\vec{a}$ to $\vec{b}$, your thumb points in the direction of $\vec{a} \times \vec{b}$.

Properties of the Cross Product

Property 1 (Anticommutativity): $\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})$

›Proof

For $\vec{b} \times \vec{a}$, the right-hand rule gives $-\hat{n}$ (the rotation is opposite), so $\vec{b} \times \vec{a} = |\vec{a}| |\vec{b}| \sin \theta \, (-\hat{n}) = - (\vec{a} \times \vec{b})$.

Property 2 (Distributivity): $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

›Proof

This follows from the geometric interpretation of the cross product as an area vector, or algebraically using components (analogous to the dot-product distributive law, but more involved).

Property 3 (Scalar multiplication): $(\lambda \vec{a}) \times \vec{b} = \lambda (\vec{a} \times \vec{b}) = \vec{a} \times (\lambda \vec{b})$

›Proof

For $\lambda \ge 0$: $(\lambda \vec{a}) \times \vec{b} = |\lambda| |\vec{a}| |\vec{b}| \sin \theta \, \hat{n} = \lambda (\vec{a} \times \vec{b})$. For $\lambda < 0$, the angle becomes $\pi - \theta$ (so $\sin(\pi-\theta) = \sin\theta$) but the right-hand rule now gives $-\hat{n}$, so $(\lambda \vec{a}) \times \vec{b} = |\lambda| |\vec{a}| |\vec{b}| \sin \theta \, (-\hat{n}) = \lambda (\vec{a} \times \vec{b})$. The same holds for $\vec{a} \times (\lambda \vec{b})$.

Property 4 (Cross product with itself): $\vec{a} \times \vec{a} = \vec{0}$

›Proof

$\vec{a} \times \vec{a} = |\vec{a}| |\vec{a}| \sin 0 \, \hat{n} = \vec{0}$. …