Scalar and Vector (Cross and Dot) Products

Introduction to Vectors

In mathematics and physics, a vector is a quantity that has both magnitude and direction. This is in contrast to a scalar, which only has magnitude. Examples of scalars include temperature, mass, and speed. Examples of vectors include displacement, velocity, and force. Vectors are typically represented by arrows, where the length of the arrow indicates the magnitude and the arrowhead indicates the direction.

In coordinate geometry, vectors can be represented using components. For instance, a vector in a 2D plane can be written as $\vec{v} = \langle v_x, v_y \rangle$ or $v_x\hat{i} + v_y\hat{j}$, where $v_x$ and $v_y$ are the components of the vector along the x-axis and y-axis, respectively. Similarly, in 3D space, a vector can be written as $\vec{v} = \langle v_x, v_y, v_z \rangle$ or $v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$, where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are the unit vectors along the x, y, and z axes, respectively.

Scalar (Dot) Product

The scalar product, also known as the dot product, is an operation that takes two vectors and returns a single scalar value. It is denoted by a dot ($\cdot$) between the two vectors.

Definition of Dot Product

If $\vec{a}$ and $\vec{b}$ are two vectors, their dot product is defined as:

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$

where $|\vec{a}|$ is the magnitude of vector $\vec{a}$, $|\vec{b}|$ is the magnitude of vector $\vec{b}$, and $\theta$ is the angle between the two vectors. The angle $\theta$ is usually taken to be in the range $[0, \pi]$.

The dot product is "scalar" because the result of the operation is a scalar quantity, not a vector.

Geometric Interpretation of Dot Product

The dot product has a significant geometric meaning. $|\vec{b}| \cos \theta$ represents the scalar projection of vector $\vec{b}$ onto the direction of vector $\vec{a}$. Similarly, $|\vec{a}| \cos \theta$ represents the scalar projection of vector $\vec{a}$ onto the direction of vector $\vec{b}$. Therefore, the dot product can be interpreted as the product of the magnitude of one vector and the scalar projection of the other vector onto the first one.

If $\vec{a} \cdot \vec{b} = 0$, and neither $\vec{a}$ nor $\vec{b}$ is a zero vector, then $\cos \theta = 0$, which implies $\theta = \frac{\pi}{2}$ (or 90 degrees). This means that the two vectors are perpendicular (orthogonal) to each other.

Properties of Dot Product

• Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
• Distributive over addition: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
• Scalar multiplication: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (k\vec{b})$, where $k$ is a scalar.
• Magnitude squared: $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Dot Product in Component Form

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then their dot product is calculated by multiplying corresponding components and summing the results:

$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

This formula is derived from the properties of dot products with unit vectors: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

Applications of Dot Product

• Finding the angle between two vectors: From the definition, $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$.
• Checking for orthogonality: If $\vec{a} \cdot \vec{b} = 0$, the vectors are orthogonal.
• Work done in physics: If a constant force $\vec{F}$ moves an object through a displacement $\vec{d}$, the work done $W$ is given by $W = \vec{F} \cdot \vec{d}$.
• Projection of one vector onto another: The scalar projection of $\vec{b}$ onto $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$. The vector projection of $\vec{b}$ onto $\vec{a}$ is $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\right)\vec{a}$.

Vector (Cross) Product

The vector product, also known as the cross product, is an operation that takes two vectors and returns a new vector. It is defined only for vectors in three-dimensional space. It is denoted by a cross ($\times$) between the two vectors.

Definition of Cross Product

If $\vec{a}$ and $\vec{b}$ are two vectors in 3D space, their cross product is defined as:

$\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \hat{n}$

where $|\vec{a}|$ is the magnitude of vector $\vec{a}$, $|\vec{b}|$ is the magnitude of vector $\vec{b}$, $\theta$ is the angle between the two vectors ($0 \le \theta \le \pi$), and $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. The direction of $\hat{n}$ is determined by the right-hand rule.

The cross product is "vector" because the result of the operation is a vector quantity.

Geometric Interpretation of Cross Product

The magnitude of the cross product, $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$, represents the area of the parallelogram formed by vectors $\vec{a}$ and $\vec{b}$ as adjacent sides.

The direction of the cross product vector $\vec{a} \times \vec{b}$ is perpendicular to the plane containing $\vec{a}$ and $\vec{b}$. The direction 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 $\vec{a} \times \vec{b}$.

If $\vec{a} \times \vec{b} = \vec{0}$ (the zero vector), and neither $\vec{a}$ nor $\vec{b}$ is a zero vector, then $\sin \theta = 0$, which implies $\theta = 0$ or $\theta = \pi$. This means that the two vectors are parallel or anti-parallel to each other.

Properties of Cross Product

• Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$. This is a crucial difference from the dot product.
• Distributive over addition: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$
• Scalar multiplication: $(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b}) = \vec{a} \times (k\vec{b})$, where $k$ is a scalar.
• Cross product with itself: $\vec{a} \times \vec{a} = \vec{0}$
• Cross product with zero vector: $\vec{a} \times \vec{0} = \vec{0}$
• Parallel vectors: If $\vec{a}$ and $\vec{b}$ are parallel, $\vec{a} \times \vec{b} = \vec{0}$.

Cross Product in Component Form

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, their cross product can be computed using a determinant of a matrix:

$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Expanding this determinant gives:

$\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}$

This formula is derived from the cross products of the unit vectors: $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}$ $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ $\hat{j} \times \hat{i} = -\hat{k}$, $\hat{k} \times \hat{j} = -\hat{i}$, $\hat{i} \times \hat{k} = -\hat{j}$

Notice the cyclic order: $\hat{i} \to \hat{j} \to \hat{k} \to \hat{i}$.

Applications of Cross Product

• Finding a vector perpendicular to two given vectors: The result of $\vec{a} \times \vec{b}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$.
• Calculating the area of a parallelogram and a triangle: The area of the parallelogram formed by $\vec{a}$ and $\vec{b}$ is $|\vec{a} \times \vec{b}|$. The area of the triangle with sides $\vec{a}$ and $\vec{b}$ is $\frac{1}{2}|\vec{a} \times \vec{b}|$.
• Torque in physics: Torque ($\vec{\tau}$) is defined as the cross product of the position vector ($\vec{r}$) and the force vector ($\vec{F}$): $\vec{\tau} = \vec{r} \times \vec{F}$.
• Angular momentum: Angular momentum ($\vec{L}$) is given by $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p}$ is the linear momentum.

Scalar Triple Product

The scalar triple product involves three vectors, say $\vec{a}$, $\vec{b}$, and $\vec{c}$. It is defined as the dot product of one vector with the cross product of the other two:

$\vec{a} \cdot (\vec{b} \times \vec{c})$

The result of the scalar triple product is a scalar.

Geometric Interpretation of Scalar Triple Product

The absolute value of the scalar triple product, $|\vec{a} \cdot (\vec{b} \times \vec{c})|$, represents the volume of the parallelepiped formed by the three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as adjacent edges.

If the scalar triple product is zero, it means the volume of the parallelepiped is zero, which implies that the three vectors are coplanar (lie in the same plane).

Scalar Triple Product in Component Form

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, and $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$, the scalar triple product can be computed as the determinant of a 3x3 matrix:

$\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$

Properties of Scalar Triple Product

• The scalar triple product is invariant under cyclic permutation: $\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b})$
• The scalar triple product changes sign under non-cyclic permutation: $\vec{a} \cdot (\vec{b} \times \vec{c}) = - \vec{a} \cdot (\vec{c} \times \vec{b})$
• The dot and cross can be interchanged without changing the value (as long as the cyclic order is maintained): $\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c}$
• If any two vectors are equal or parallel, the scalar triple product is zero.

Vector Triple Product

The vector triple product involves three vectors, say $\vec{a}$, $\vec{b}$, and $\vec{c}$. It is defined as the cross product of one vector with the cross product of the other two:

$\vec{a} \times (\vec{b} \times \vec{c})$

The result of the vector triple product is a vector.

Vector Triple Product Identity

The vector triple product can be expanded using the following identity:

$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$

This identity is extremely useful as it expresses the vector triple product in terms of dot products and vector scaling, which are generally easier to compute.

Note that the vector triple product is NOT associative: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ In fact, $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$, which is different from the above.

Properties of Vector Triple Product

• The result vector $\vec{a} \times (\vec{b} \times \vec{c})$ is coplanar with $\vec{b}$ and $\vec{c}$.
• It is important to correctly identify which vector is "outside" the cross product in the expansion formula.

Summary Table of Products

Here is a table summarizing the key features of scalar and vector products:

Example Problems

Example 1: Dot Product

Find the angle between vectors $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$.

Solution:

First, calculate the dot product: $\vec{a} \cdot \vec{b} = (2)(1) + (1)(-1) + (-1)(2) = 2 - 1 - 2 = -1$.

Next, calculate the magnitudes: $|\vec{a}| = \sqrt{2^2 + 1^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$. $|\vec{b}| = \sqrt{1^2 + (-1)^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$.

Now, use the formula $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$: $\cos \theta = \frac{-1}{\sqrt{6} \cdot \sqrt{6}} = \frac{-1}{6}$.

Therefore, the angle $\theta = \arccos\left(-\frac{1}{6}\right)$.

Example 2: Cross Product

Find the vector $\vec{c} = \vec{a} \times \vec{b}$, where $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$.

Solution:

Using the determinant formula: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 2 & -1 & 1 \end{vmatrix}$

$= \hat{i}((2)(1) - (1)(-1)) - \hat{j}((1)(1) - (1)(2)) + \hat{k}((1)(-1) - (2)(2))$ $= \hat{i}(2 - (-1)) - \hat{j}(1 - 2) + \hat{k}(-1 - 4)$ $= \hat{i}(3) - \hat{j}(-1) + \hat{k}(-5)$ $= 3\hat{i} + \hat{j} - 5\hat{k}$.

So, $\vec{c} = 3\hat{i} + \hat{j} - 5\hat{k}$.

Example 3: Scalar Triple Product

Find the volume of the parallelepiped formed by vectors $\vec{a} = \hat{i} + \hat{j}$, $\vec{b} = \hat{j} + \hat{k}$, and $\vec{c} = \hat{k} + \hat{i}$.

Solution:

The volume is given by $|\vec{a} \cdot (\vec{b} \times \vec{c})|$. First, write the vectors in component form: $\vec{a} = \langle 1, 1, 0 \rangle$, $\vec{b} = \langle 0, 1, 1 \rangle$, $\vec{c} = \langle 1, 0, 1 \rangle$.

Calculate the scalar triple product using the determinant: $\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{vmatrix}$ $= 1((1)(1) - (1)(0)) - 1((0)(1) - (1)(1)) + 0((0)(0) - (1)(1))$ $= 1(1 - 0) - 1(0 - 1) + 0$ $= 1(1) - 1(-1) = 1 + 1 = 2$.

The volume of the parallelepiped is $|2| = 2$ cubic units.

Example 4: Vector Triple Product

Evaluate $\vec{a} \times (\vec{b} \times \vec{c})$ for $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j}$, and $\vec{c} = \hat{j} + \hat{k}$.

Solution:

Using the identity $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$:

Calculate the dot products: $\vec{a} \cdot \vec{c} = (1)(0) + (1)(1) + (1)(1) = 0 + 1 + 1 = 2$. $\vec{a} \cdot \vec{b} = (1)(2) + (1)(-1) + (1)(0) = 2 - 1 + 0 = 1$.

Substitute these values into the identity: $\vec{a} \times (\vec{b} \times \vec{c}) = (2)\vec{b} - (1)\vec{c}$ $= 2(2\hat{i} - \hat{j}) - 1(\hat{j} + \hat{k})$ $= (4\hat{i} - 2\hat{j}) - (\hat{j} + \hat{k})$ $= 4\hat{i} - 2\hat{j} - \hat{j} - \hat{k}$ $= 4\hat{i} - 3\hat{j} - \hat{k}$.