3.3 Cross Product and Vector Direction | Vectors | Mathematics 11

Cross Product (Vector Product)

The cross product (or vector product) of two vectors $a$ and $b$ is defined as:

$a \times b = ∣ a ∣∣ b ∣ sin θ \hat{n}$

where $θ$ is the angle between $a$ and $b$ , and $\hat{n}$ is the unit vector perpendicular to both, determined by the right-hand rule.

Component Form

If $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ and $b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ , then:

$a \times b = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}$

$= (a_{2} b_{3} - a_{3} b_{2}) \hat{i} - (a_{1} b_{3} - a_{3} b_{1}) \hat{j} + (a_{1} b_{2} - a_{2} b_{1}) \hat{k}$

Key Properties of Cross Product

Property	Statement
Anti-commutativity	$a \times b = - (b \times a)$
Distributive law	$a \times (b + c) = a \times b + a \times c$
Scalar multiplication	$(k a) \times b = k (a \times b)$
Parallel vectors	$a \times b = 0$ if $a ∥ b$
Self cross product	$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
Cyclic rule	$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$

Geometric Interpretation

$∣ a \times b ∣$ = area of the parallelogram with sides $a$ and $b$
Area of triangle with sides $a$ and $b$ : $A_{△} = \frac{1}{2} ∣ a \times b ∣$

Finding the Angle Between Two Vectors

From the definition of cross product:

$sin θ = \frac{∣ a \times b ∣}{∣ a ∣∣ b ∣}$

Unit Vector Perpendicular to Two Vectors

$\hat{n} = \pm \frac{a \times b}{∣ a \times b ∣}$

Lagrange Identity

$∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$

Proof: Using $sin^{2} θ + cos^{2} θ = 1$ : $∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} sin^{2} θ = ∣ a ∣^{2} ∣ b ∣^{2} (1 - cos^{2} θ) = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$

Scalar Triple Product

The scalar triple product of three vectors $a$ , $b$ , $c$ is:

$[a, b, c] = a \cdot (b \times c)$

The result is a scalar.

Determinant Form

If $a = (a_{1}, a_{2}, a_{3})$ , $b = (b_{1}, b_{2}, b_{3})$ , $c = (c_{1}, c_{2}, c_{3})$ :

$a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$

Interchangeability of Dot and Cross

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

Dot and cross can be interchanged in the scalar triple product without changing the value.

Proof: Both sides equal the same $3 \times 3$ determinant with rows $a$ , $b$ , $c$ .

Cyclic Property

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

Volume of a Parallelepiped

The volume of the parallelepiped determined by vectors $a$ , $b$ , $c$ is:

$V_{parallelepiped} = ∣ a \cdot (b \times c) ∣$

Volume of a Tetrahedron

$V_{tetrahedron} = \frac{1}{6} ∣ a \cdot (b \times c) ∣$

Example: Find the volume of the parallelepiped with edges $a = \hat{i} + 2 \hat{j} + 3 \hat{k}$ , $b = 2 \hat{i} - \hat{j} + \hat{k}$ , $c = \hat{i} + \hat{j} - \hat{k}$ .

$V = 121 2 - 1 1 31 - 1$

$= ∣1 ((- 1) (- 1) - (1) (1)) - 2 ((2) (- 1) - (1) (1)) + 3 ((2) (1) - (- 1) (1)) ∣$ $= ∣1 (1 - 1) - 2 (- 2 - 1) + 3 (2 + 1) ∣$ $= ∣0 + 6 + 9∣ = 15 cubic units$

Coplanar Vectors

Three vectors $a$ , $b$ , $c$ are coplanar if they lie in the same plane.

Condition for Coplanarity

$a \cdot (b \times c) = 0$

Equivalently: $a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} = 0$

Reason: Coplanar vectors form a parallelepiped of zero volume, so the scalar triple product must be zero.

Example: Show that $a = \hat{i} + \hat{j}$ , $b = \hat{j} + \hat{k}$ , $c = \hat{i} + \hat{k}$ are coplanar.

$101110011 = 1 (1 - 0) - 1 (0 - 1) + 0 = 1 + 1 = 2 \neq = 0$

These vectors are not coplanar.

Result: Cross Products When $a + b + c = 0$

If $a + b + c = 0$ , then:

$a \times b = b \times c = c \times a$

Proof: From $c = - (a + b)$ , cross both sides with $a$ : $a \times c = a \times (- (a + b)) = - (a \times a) - (a \times b) = - a \times b$ $∴ c \times a = a \times b$

Similarly, crossing with $b$ gives $a \times b = b \times c$ .

Cross Product (Vector Product)

The cross product (or vector product) of two vectors $a$ and $b$ is defined as:

$a \times b = ∣ a ∣∣ b ∣ sin θ \hat{n}$

where $θ$ is the angle between $a$ and $b$ , and $\hat{n}$ is the unit vector perpendicular to both, determined by the right-hand rule.

Component Form

If $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ and $b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ , then:

$a \times b = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}$

$= (a_{2} b_{3} - a_{3} b_{2}) \hat{i} - (a_{1} b_{3} - a_{3} b_{1}) \hat{j} + (a_{1} b_{2} - a_{2} b_{1}) \hat{k}$

Key Properties of Cross Product

Property	Statement
Anti-commutativity	$a \times b = - (b \times a)$
Distributive law	$a \times (b + c) = a \times b + a \times c$
Scalar multiplication	$(k a) \times b = k (a \times b)$
Parallel vectors	$a \times b = 0$ if $a ∥ b$
Self cross product	$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
Cyclic rule	$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$

Geometric Interpretation

$∣ a \times b ∣$ = area of the parallelogram with sides $a$ and $b$
Area of triangle with sides $a$ and $b$ : $A_{△} = \frac{1}{2} ∣ a \times b ∣$

Finding the Angle Between Two Vectors

From the definition of cross product:

$sin θ = \frac{∣ a \times b ∣}{∣ a ∣∣ b ∣}$

Unit Vector Perpendicular to Two Vectors

$\hat{n} = \pm \frac{a \times b}{∣ a \times b ∣}$

Lagrange Identity

$∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$

Scalar Triple Product

The scalar triple product of three vectors $a$ , $b$ , $c$ is:

$[a, b, c] = a \cdot (b \times c)$

The result is a scalar.

Determinant Form

If $a = (a_{1}, a_{2}, a_{3})$ , $b = (b_{1}, b_{2}, b_{3})$ , $c = (c_{1}, c_{2}, c_{3})$ :

$a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$

Interchangeability of Dot and Cross

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

Dot and cross can be interchanged in the scalar triple product without changing the value.

Proof: Both sides equal the same $3 \times 3$ determinant with rows $a$ , $b$ , $c$ .

Cyclic Property

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

Volume of a Parallelepiped

The volume of the parallelepiped determined by vectors $a$ , $b$ , $c$ is:

$V_{parallelepiped} = ∣ a \cdot (b \times c) ∣$

Volume of a Tetrahedron

$V_{tetrahedron} = \frac{1}{6} ∣ a \cdot (b \times c) ∣$

Example: Find the volume of the parallelepiped with edges $a = \hat{i} + 2 \hat{j} + 3 \hat{k}$ , $b = 2 \hat{i} - \hat{j} + \hat{k}$ , $c = \hat{i} + \hat{j} - \hat{k}$ .

$V = 121 2 - 1 1 31 - 1$

$= ∣1 ((- 1) (- 1) - (1) (1)) - 2 ((2) (- 1) - (1) (1)) + 3 ((2) (1) - (- 1) (1)) ∣$ $= ∣1 (1 - 1) - 2 (- 2 - 1) + 3 (2 + 1) ∣$ $= ∣0 + 6 + 9∣ = 15 cubic units$

Coplanar Vectors

Three vectors $a$ , $b$ , $c$ are coplanar if they lie in the same plane.

Condition for Coplanarity

$a \cdot (b \times c) = 0$

Equivalently: $a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} = 0$

Reason: Coplanar vectors form a parallelepiped of zero volume, so the scalar triple product must be zero.

Example: Show that $a = \hat{i} + \hat{j}$ , $b = \hat{j} + \hat{k}$ , $c = \hat{i} + \hat{k}$ are coplanar.

$101110011 = 1 (1 - 0) - 1 (0 - 1) + 0 = 1 + 1 = 2 \neq = 0$

These vectors are not coplanar.

Result: Cross Products When $a + b + c = 0$

If $a + b + c = 0$ , then:

$a \times b = b \times c = c \times a$

Proof: From $c = - (a + b)$ , cross both sides with $a$ : $a \times c = a \times (- (a + b)) = - (a \times a) - (a \times b) = - a \times b$ $∴ c \times a = a \times b$

Similarly, crossing with $b$ gives $a \times b = b \times c$ .