Definition of Scalar Product
The scalar product, also known as the dot product, is a fundamental operation in vector algebra. For two vectors $\mathbf{a} = \begin{pmatrix} a_1\\a_2 \\a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1\\b_2 \\b_3 \end{pmatrix}$ in three-dimensional space, their scalar product is defined as:
$$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$
This operation takes two vectors as input and produces a scalar (hence the name) as output.
NoteThe scalar product is denoted by a dot (·) between the vectors, which is why it's also called the dot product.
Properties of the Scalar Product
The scalar product possesses several important properties that make it a versatile tool in vector calculations:
- Commutativity: $\mathbf{v} \cdot \mathbf{w} = \mathbf{w} \cdot \mathbf{v}$ This means the order of the vectors doesn't matter when calculating their scalar product.
- Distributivity over addition: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$ This property allows us to break down complex scalar products into simpler ones.
- Scalar multiplication: $(k\mathbf{v}) \cdot \mathbf{w} = k(\mathbf{v} \cdot \mathbf{w})$ Where $k$ is a scalar. This property shows how scalar multiplication interacts with the dot product.
- Self-dot product: $\mathbf{v} \cdot \mathbf{v} = ||\mathbf{v}||^2$ The scalar product of a vector with itself equals the square of its magnitude. This is because
$$\mathbf{v}\cdot\mathbf{v} = v_1^2 + v_2^2 + v_3^2 = \sqrt{v_1^2 + v_2^2 + v_3^2}^2 = ||\mathbf{v}||^2$$
ExampleLet $\mathbf{a} = \begin{pmatrix}1 \\2 \\3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4\\ 5\\6 \end{pmatrix}$. Then:
$\mathbf{a} \cdot \mathbf{b} = 1(4) + 2(5) + 3(6) = 4 + 10 + 18 = 32$
Demonstrating commutativity: $\mathbf{b} \cdot \mathbf{a} = 4(1) + 5(2) + 6(3) = 4 + 10 + 18 = 32$
Geometric Interpretation
One of the most powerful aspects of the scalar product is its geometric interpretation. For two vectors $\mathbf{v}$ and $\mathbf{w}$, their scalar product is related to their magnitudes and the angle between them by the formula:
$$\mathbf{v} \cdot \mathbf{w} = ||\mathbf{v}||\times ||\mathbf{w}||\cos\theta$$
Where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$, and $||\mathbf{v}||$ and $||\mathbf{w}||$ are the magnitudes of the vectors.
This formula provides a powerful way to find angles between vectors and has numerous applications in physics, particularly in calculating work done by a force.
The above figure shows a quick geometrical interpretation of the dot product. If you have two vectors, and draw a perpendicular line with one attaching it to another to make a right angle triangle, the length of the the 'adjacent' component is equal to $||\mathbf{v}||\cos(\theta)$ where $\mathbf{v}$ is the hypotenuse of this triangle.
By multiplying this length by $\mathbf{w}$ which is going to be the vector pointing in the same direction as the adjacent component of the right angle triangle, we will get $||\mathbf{w}||\,||\mathbf{v}||\cos\theta$
In physics, this can be used to represent the force in a direction from each component $x,y,z$, or how much 'power' one vector has combined with another in a particular direction.
Applications of the Dot Product
Perpendicular Vectors
One key application of the dot product is in determining whether two vectors are perpendicular (orthogonal). Two non-zero vectors $\mathbf{v}$ and $\mathbf{w}$ are perpendicular if and only if their scalar product is zero:
$$\mathbf{v} \cdot \mathbf{w} = 0 \iff \mathbf{v} \perp \mathbf{w}$$
This follows from the geometric interpretation, as $\cos 90° = 0$.
ExampleLet $\mathbf{a} = \begin{pmatrix} 1\\2 \\-1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2\\-1 \\ 2\end{pmatrix}$. Are they perpendicular?
$\mathbf{a} \cdot \mathbf{b} = 1(2) + 2(-1) + (-1)(2) = 2 - 2 - 2 = -2$
Since $\mathbf{a} \cdot \mathbf{b} \neq 0$, the vectors are not perpendicular.
Parallel Vectors
For parallel vectors, the angle between them is either 0° (same direction) or 180° (opposite directions). In either case, $|\cos\theta| = 1$, which leads to:
$$||\mathbf{v} \cdot \mathbf{w}|| = ||\mathbf{v}||\,||\mathbf{w}||$$
This equality holds if and only if the vectors are parallel.
Finding Angles Between Vectors
The scalar product formula can be rearranged to find the angle between two vectors:
$$\theta = \arccos\left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}||\mathbf{w}|}\right)$$
This is particularly useful in geometry and physics problems where angles between vectors are required.
ExampleFind the angle between vectors $\mathbf{a} = \begin{pmatrix} 1\\1 \\1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1\\ 0\\-1 \end{pmatrix}$.
First, calculate the scalar product: $\mathbf{a} \cdot \mathbf{b} = 1(1) + 1(0) + 1(-1) = 0$
Then, calculate the magnitudes: $||\mathbf{a}|| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$
$||\mathbf{b}|| = \sqrt{1^2 + 0^2 + (-1)^2} = \sqrt{2}$
Now, use the formula: $\theta = \arccos\left(\frac{0}{\sqrt{3}\sqrt{2}}\right) = \arccos(0) = 90°$
The vectors are perpendicular, which we could have deduced directly from $\mathbf{a} \cdot \mathbf{b} = 0$.
Vector Product in Mathematics
The vector product, also known as the cross product, is a fundamental operation in vector algebra that produces a vector perpendicular to both of the input vectors.
NoteThis concept is crucial in various fields of mathematics, physics, and engineering.
Definition and Formula
The vector product of two vectors $\mathbf{v}$ and $\mathbf{w}$ is defined as:
$$ \mathbf{v} \times \mathbf{w} = |\mathbf{v}||\mathbf{w}| \sin \theta \mathbf{n} $$
Where:
- $|\mathbf{v}|$ and $|\mathbf{w}|$ are the magnitudes of the vectors
- $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$
- $\mathbf{n}$ is the unit normal vector perpendicular to both $\mathbf{v}$ and $\mathbf{w}$
The direction of $\mathbf{n}$ is determined by the right-hand screw rule: if you curl the fingers of your right hand from $\mathbf{v}$ to $\mathbf{w}$ through the smaller angle, your thumb points in the direction of $\mathbf{n}$.
Properties of Vector Product
The vector product possesses several important properties:
- Anticommutativity: $\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})$ This means that changing the order of the vectors in a cross product negates the result.
- Distributivity over addition: $\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = \mathbf{u} \times \mathbf{v} + \mathbf{u} \times \mathbf{w}$ This property allows us to break down complex cross products into simpler ones.
- Scalar multiplication: $(k\mathbf{v}) \times \mathbf{w} = k(\mathbf{v} \times \mathbf{w})$ Where $k$ is a scalar.
- Self-cross product: $\mathbf{v} \times \mathbf{v} = \mathbf{0}$ The cross product of a vector with itself is always the zero vector.
- Parallel vectors: For non-zero vectors, $\mathbf{v} \times \mathbf{w} = \mathbf{0}$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel.
Let $\mathbf{v} = (1, 2, 3)$ and $\mathbf{w} = (4, 5, 6)$. Their cross product is:
$\mathbf{v} \times \mathbf{w} = (2 \cdot 6 - 3 \cdot 5, 3 \cdot 4 - 1 \cdot 6, 1 \cdot 5 - 2 \cdot 4) = (-3, 6, -3)$
You can verify that this result is perpendicular to both $\mathbf{v}$ and $\mathbf{w}$ by computing the dot products:
$\mathbf{v} \cdot (\mathbf{v} \times \mathbf{w}) = 1(-3) + 2(6) + 3(-3) = 0$ $\mathbf{w} \cdot (\mathbf{v} \times \mathbf{w}) = 4(-3) + 5(6) + 6(-3) = 0$
Geometric Interpretation
- The magnitude of the vector product, $|\mathbf{v} \times \mathbf{w}|$, has a significant geometric interpretation.
- It represents the area of the parallelogram formed by the two vectors $\mathbf{v}$ and $\mathbf{w}$.
- This property makes the vector product extremely useful in calculating areas.
To find the area of a triangle with vertices $A(1, 0, 0)$, $B(0, 2, 0)$, and $C(0, 0, 3)$:
- Form vectors $\mathbf{AB} = (-1, 2, 0)$ and $\mathbf{AC} = (-1, 0, 3)$
- Calculate the cross product: $\mathbf{AB} \times \mathbf{AC} = (6, 3, -2)$
- Find the magnitude: $|\mathbf{AB} \times \mathbf{AC}| = \sqrt{6^2 + 3^2 + (-2)^2} = 7$
- The area of the triangle is half this magnitude: $\frac{7}{2} = 3.5$ square units
The cross product is a powerful tool when finding the shortest distance between objects, such as lines or planes in three-dimensional space.
Why? The shortest distance vector must be perpendicular to both objects. Since the cross product of two vectors produces a vector that is perpendicular to both, it helps define the direction of the shortest distance.
Applications in Physics
The vector product plays a crucial role in various physics concepts, particularly in electromagnetism and mechanics.
- Magnetic Force: The force $\mathbf{F}$ on a charged particle moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ is given by: $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$ Where $q$ is the charge of the particle.
- Torque: The torque $\boldsymbol{\tau}$ exerted by a force $\mathbf{F}$ applied at a position $\mathbf{r}$ relative to a pivot point is: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$
When solving problems involving magnetic forces or torques, always pay attention to the direction of the resulting vector, as it's perpendicular to both input vectors.
Common MistakeStudents often confuse the dot product and cross product. Remember:
- The dot product results in a scalar.
- The cross product results in a vector.
- The dot product is commutative ($\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$), while the cross product is anti-commutative ($\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$).
Consider how the certainty we have in mathematical operations like the cross product compares to the certainty in other areas of knowledge, such as experimental sciences or human sciences.
