Lecture 4: Vector and Triple Products

Vector Product (Cross Product) Overview

Definition of Vector Product

• Scalar Product Recap: A scalar product (dot product) combines two vectors to produce a scalar (Vector $\cdot$ Vector $\Rightarrow$ Scalar).

• Vector Product (Cross Product) Definition: A vector product (cross product) combines two vectors to produce a vector (Vector $\times$ Vector $\Rightarrow$ Vector).

• Formula: The vector product of two vectors $a$ and $b$ is given by: $a \times b = |a| |b|\sin \theta \hat{e}$ where:

  • $|a|$ is the magnitude of vector $a$.

  • $|b|$ is the magnitude of vector $b$.

  • $\theta$ is the angle between vectors $a$ and $b$ ($0 \le \theta \le \pi$).

  • $\hat{e}$ is a unit vector perpendicular to both $a$ and $b$.

• Direction of $\hat{e}$: The direction of $\hat{e}$ is determined by the right-hand rule. If you curl the fingers of your right hand from vector $a$ to vector $b$ through the angle $\theta$, your thumb points in the direction of $\hat{e}$. The illustration of the right-hand rule demonstrates this concept.

• Magnitude and Direction: The vector $a \times b$ has a magnitude of $|a| |b|\sin \theta$ and a direction given by $\hat{e}$.

Vector Product for Parallel Vectors

• General Formula: $a \times b = |a| |b|\sin \theta \hat{e}$

• Zero Vector Condition: If either vector $a$ or vector $b$ is the zero vector ($a = 0$ or $b = 0$), then their cross product is the zero vector ($a \times b = 0$).

• Parallel Vector Condition: If vectors $a$ and $b$ are parallel, the angle between them is either $0$ (same direction) or $\pi$ (opposite directions). In both cases, $\sin \theta = 0$. Therefore, if $a$ and $b$ are parallel, then $a \times b = 0$.

• Special Case: The cross product of a vector with itself is always the zero vector ($a \times a = 0$) since a vector is parallel to itself.

• Undefined Unit Vector: The unit vector $\hat{e}$ is not defined if $a$ and $b$ are parallel or one of them is a zero vector. Despite this, the cross product $a \times b$ is still well-defined as the zero vector ($0$).

• Converse: If $a \times b = 0$ and both $a$ and $b$ are non-zero vectors, it implies that $\sin \theta = 0$. This means $\theta = 0$ or $\theta = \pi$, which indicates that vectors $a$ and $b$ are parallel.

• Diagnostic Tool: The scalar product is used to test for perpendicular vectors ($a \cdot b = 0$). Conversely, the vector product is used to test for parallel vectors ($a \times b = 0$).

Vector Product Properties

Let $a$, $b$, $c$ be vectors and $\lambda$ be a scalar:

• Anticommutative Property: The order of the vectors in a cross product matters. Swapping the order reverses the direction of the resulting vector. $b \times a = - (a \times b)$

  • Explanation: When $a \times b$ is calculated, $\hat{e}$ points in a specific direction. For $b \times a$, the right-hand rule dictates that $\hat{e}$ will point in the opposite direction ($-\hat{e}$).

• Distributive over Vector Addition: The cross product is distributive over vector addition.

  • $a \times (b + c) = (a \times b) + (a \times c)$

  • $(a + b) \times c = (a \times c) + (b \times c)$

• Associative over Scalar Multiplication: A scalar factor can be factored out or in with any of the vectors.
  $(\lambda a) \times b = \lambda (a \times b) = a \times (\lambda b)$

• Parallel Condition (revisited): If $a$ and $b$ are non-zero vectors, then $a \times b = 0$ if and only if $a$ and $b$ are parallel. As a specific instance, $a \times a = 0$.

• Non-Associative in General: The vector product is generally not associative for three vectors.
  $a \times (b \times c) \neq (a \times b) \times c$

Area of a Parallelogram

• The magnitude of the vector product of two vectors $a$ and $b$ (originating from the same point) represents the area of the parallelogram formed by these two vectors as adjacent sides.

• Formula: $Area = |a \times b| = |a| |b| \sin \theta$

• Geometric Derivation: The area of the parallelogram OACB is twice the area of triangle OAB. The area of triangle OAB is $\frac{1}{2} \cdot |OA| \cdot h$, where $h$ is the perpendicular height from B to OA. The height $h = |OB| \sin \theta = |b| \sin \theta$. So, the area of triangle OAB is $\frac{1}{2} |a| |b| \sin \theta$. Therefore, the area of the parallelogram is $2 \cdot \frac{1}{2} |a| |b| \sin \theta = |a| |b| \sin \theta$. This is precisely the magnitude of their cross product.

Vector Product in Cartesian Form

• Cross Products of Cartesian Unit Vectors:

  • $i \times i = 0$, $j \times j = 0$, $k \times k = 0$

  • $i \times j = k$

  • $j \times k = i$

  • $k \times i = j$

  • $j \times i = -k$

  • $k \times j = -i$

  • $i \times k = -j$

• For any two vectors in Cartesian coordinates:
  Let $a = a<em>1 i + a</em>2 j + a<em>3 k$ And $b = b</em>1 i + b<em>2 j + b</em>3 k$

• Expanded Calculation: Performing the distributive property with the unit vector cross products:
  $a \times b = (a<em>1 i + a</em>2 j + a<em>3 k) \times (b</em>1 i + b<em>2 j + b</em>3 k)$
  $= a<em>1 b</em>2 (i \times j) + a<em>1 b</em>3 (i \times k) + a<em>2 b</em>1 (j \times i) + a<em>2 b</em>3 (j \times k) + a<em>3 b</em>1 (k \times i) + a<em>3 b</em>2 (k \times j)$
  $= a<em>1 b</em>2 k + a<em>1 b</em>3 (-j) + a<em>2 b</em>1 (-k) + a<em>2 b</em>3 i + a<em>3 b</em>1 j + a<em>3 b</em>2 (-i)$
  $= (a<em>2 b</em>3 - a<em>3 b</em>2) i + (a<em>3 b</em>1 - a<em>1 b</em>3) j + (a<em>1 b</em>2 - a<em>2 b</em>1) k$

• Determinant Form (Easier to Remember): The vector product can be concisely calculated using the determinant of a $3 \times 3$ matrix:
  $a \times b = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k} \ a<em>1 &amp; a</em>2 &amp; a<em>3 \ b</em>1 &amp; b<em>2 &amp; b</em>3 \end{vmatrix}$
  Expanding the determinant:
  $= \hat{i} (a<em>2 b</em>3 - a<em>3 b</em>2) - \hat{j} (a<em>1 b</em>3 - a<em>3 b</em>1) + \hat{k} (a<em>1 b</em>2 - a<em>2 b</em>1)$
  $= (a<em>2 b</em>3 - a<em>3 b</em>2) \hat{i} + (a<em>3 b</em>1 - a<em>1 b</em>3) \hat{j} + (a<em>1 b</em>2 - a<em>2 b</em>1) \hat{k}$

Examples of Vector Product Calculation

• Example 1: Given vectors $a = 6i + 2j - 3k$ and $b = 4i - j - 2k$.
  Find $a \times b$.
  $a \times b = \begin{vmatrix} i &amp; j &amp; k \ 6 &amp; 2 &amp; -3 \ 4 &amp; -1 &amp; -2 \end{vmatrix}$
  $= i((2)(-2) - (-3)(-1)) - j((6)(-2) - (-3)(4)) + k((6)(-1) - (2)(4))$
  $= i(-4 - 3) - j(-12 + 12) + k(-6 - 8)$
  $= -7i - 0j - 14k$
  $= -7i - 14k$

• Example 2: Find a vector perpendicular to $a = 4i - 2j + 3k$ and $b = 5i + j - 4k$.
  The cross product $a \times b$ gives a vector perpendicular to both $a$ and $b$.
  $a \times b = \begin{vmatrix} i &amp; j &amp; k \ 4 &amp; -2 &amp; 3 \ 5 &amp; 1 &amp; -4 \end{vmatrix}$
  $= i((-2)(-4) - (3)(1)) - j((4)(-4) - (3)(5)) + k((4)(1) - (-2)(5))$
  $= i(8 - 3) - j(-16 - 15) + k(4 + 10)$
  $= 5i - (-31)j + 14k$
  $= 5i + 31j + 14k$

Triple Products

• Triple products are operations that combine three vectors using a combination of scalar and vector products.

Scalar Triple Product

• Definition: The scalar triple product is defined as $a \cdot (b \times c)$. The result is a scalar.

• Properties (Cyclic Permutation): The scalar triple product remains unchanged under cyclic permutation of the vectors.
  $a \cdot (b \times c) = c \cdot (a \times b) = b \cdot (c \times a)$

• Determinant Form: The scalar triple product can be calculated as the determinant of a $3 \times 3$ matrix where the rows (or columns) are the components of the three vectors. $a \cdot (b \times c) = \begin{vmatrix} a<em>1 &amp; a</em>2 &amp; a<em>3 \ b</em>1 &amp; b<em>2 &amp; b</em>3 \ c<em>1 &amp; c</em>2 &amp; c_3 \end{vmatrix}$

  • This can be proven by using the determinant definition of the vector product and then performing the dot product.

• Geometric Interpretation (Volume of Parallelepiped):

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

  • The area of the base parallelogram formed by vectors $b$ and $c$ is $|b \times c|$.

  • The height of the parallelepiped relative to this base is the projection of vector $a$ onto the normal of the base, which is $|a| |\cos \phi|$, where $\phi$ is the angle between $a$ and $(b \times c)$.

  • Volume $= (Area of Base) \times (Height) = |b \times c| |a| |\cos \phi| = |a \cdot (b \times c)|$.

• Coplanarity Condition: If the scalar triple product $a \cdot (b \times c) = 0$, then the three vectors $a$, $b$, and $c$ are coplanar (lie in the same plane). This also implies they are linearly dependent, meaning one vector can be expressed as a linear combination of the other two (e.g., $a = \lambda b + \mu c$ for some scalars $\lambda$ and $\mu$).

Vector Triple Product

• Definition: The vector triple product is defined as $a \times (b \times c)$. The result is a vector.

• Non-Associative (revisited): As noted previously, the vector triple product is not associative; meaning $(a \times b) \times c \neq a \times (b \times c)$.

• Expansion Formula (BAC-CAB Rule): The vector triple product can be expanded using a formula known as the BAC-CAB rule:

  • $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$

  • $(a \times b) \times c = (a \cdot c) b - (b \cdot c) a$