Vector Algebra : Cross Product: Component Form

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Cross Product: Component Form

what you'll learn...

Overview

Vector Cross Product: Component Form

$\vec{p} \times \vec{q} = | \begin{matrix} i & j & k \\ p_{x} & p_{y} & p_{z} \\ q_{x} & q_{y} & q_{z} \end{matrix} |$ $vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z) |$

$= (p_{y} q_{z} - p_{z} q_{y}) i$ $=(p_yq_z-p_zq_y)i$ $+ (p_{z} q_{x} - p_{x} q_{z}) j$ $+ (p_zq_x-p_xq_z)j$ $+ (p_{x} q_{y} - p_{y} q_{x}) k$ $+ (p_xq_y-p_yq_x)k$

in component form

Given $\vec{p} = p_{x} i + p_{y} j + p_{z} k$ $vec p = p_x i + p_yj+p_zk$ and $\vec{q} = q_{x} i + q_{y} j + q_{z} k$ $vec q =q_x i + q_yj+q_zk$ and $| \vec{p} \times \vec{q} | = | p | | q | \sin θ$ $|vec p xx vec q| = |p||q|sin theta$
How to find $\sin θ$ $sin theta$ , or the cross product in the component form?

Component form of vector cross product : geometrical proof

$| \vec{p} \times \vec{q} |$ $|vec p xx vec q|$
$= | p | | q | \sin θ$ $quad quad = |p||q|sin theta$
$= | p | | q | \sqrt{1 - \cos^{2} θ}$ $quad quad = |p||q|sqrt(1-cos^2 theta)$
$= \sqrt{{| p |}^{2} {| q |}^{2} - {| p |}^{2} {| q |}^{2} \cos^{2} θ}$ $quad quad = sqrt(|p|^2|q|^2-|p|^2|q|^2cos^2 theta)$
$= \sqrt{{| p |}^{2} {| q |}^{2} - {| p \cdot q |}^{2}}$ $quad quad = sqrt(|p|^2|q|^2-|p cdot q|^2)$

Substitute the following
${| \vec{p} |}^{2} = (p_{x}^{2} + p_{y}^{2} + p_{z}^{2})$ $|vec p|^2 = (p_x^2+p_y^2+p_z^2)$
${| \vec{q} |}^{2} = (q_{x}^{2} + q_{y}^{2} + q_{z}^{2})$ $|vec q|^2 = (q_x^2+q_y^2+q_z^2)$
${| \vec{p} \cdot \vec{q} |}^{2} = {(p_{x} q_{x} + p_{y} q_{y} + p_{z} q_{z})}^{2}$ $|vec p cdot vec q|^2 = (p_xq_x+p_yq_y+p_zq_z)^2$
and after some algebraic rearrangement, the following is arrived at

$| \vec{p} \times \vec{q} |$ $|vec p xx vec q|$
$= \sqrt{{(p_{y} q_{z} - p_{z} q_{y})}^{2}}$ $quad quad = sqrt((p_yq_z-p_zq_y)^2)$
$\bar{+ {(p_{z} q_{x} - p_{x} q_{z})}^{2}}$ $quad quad quad quad bar(+(p_zq_x-p_xq_z)^2)$
$\bar{+ {(p_{x} q_{y} - p_{y} q_{x})}^{2}}$ $quad quad quad quad bar(+(p_xq_y-p_yq_x)^2)$
$= ∣ (p_{y} q_{z} - p_{z} q_{y}) i$ $quad quad = | (p_yq_z-p_zq_y)i$
$+ (p_{z} q_{x} - p_{x} q_{z}) j$ $quad quad quad quad + (p_zq_x-p_xq_z)j$
$+ (p_{x} q_{y} - p_{y} q_{x}) k ∣$ $quad quad quad quad + (p_xq_y-p_yq_x)k |$

It is proven that
$\vec{p} \times \vec{q}$ $vec p xx vec q$
$= (p_{y} q_{z} - p_{z} q_{y}) i$ $quad quad = (p_yq_z-p_zq_y)i$
$+ (p_{z} q_{x} - p_{x} q_{z}) j$ $quad quad quad quad + (p_zq_x-p_xq_z)j$
$+ (p_{x} q_{y} - p_{y} q_{x}) k$ $quad quad quad quad + (p_xq_y-p_yq_x)k$

The same can be written in the determinant form
$\vec{p} \times \vec{q} = | \begin{matrix} i & j & k \\ p_{x} & p_{y} & p_{z} \\ q_{x} & q_{y} & q_{z} \end{matrix} |$ $vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|$

easier proof

The proof given above is complex. A simpler proof, that a student can easily derive, is given below.

Bilinear Property: For any vector $\vec{p}, \vec{q}, \vec{r} \in V$ $vec p, vec q, vec r in bbb V$ and $λ \in ℝ$ $lambda in RR$
$(λ \vec{p} + \vec{q}) \times \vec{r}$ $(lambda vec p + vec q) xx vec r$ $= λ (\vec{p} \times \vec{r}) + (\vec{q} \times \vec{r})$ $= lambda (vec p xx vec r) + (vec q xx vec r)$
This is explained and proven in properties of the cross product. For now, consider this to be true.

A vector $\vec{p} = p_{x} i + p_{y} j + p_{z} k$ $vec p = p_x i + p_y j + p_z k$ is sum of scalar multiple of vectors. $i, j, k$ $i, j, k$ are unit vectors, and the scalar multiples are $p_{x}, p_{y}, p_{z}$ $p_x, p_y, p_z$ .

The same applies to $\vec{q} = q_{x} i + q_{y} j + q_{z} k$ $vec q = q_x i + q_y j + q_z k$ -- sum of multiple of vectors.

Proof for component form of vector cross product using bilinear property of cross product.

$\vec{p} \times \vec{q}$ $vec p xx vec q$
$= (p_{x} i + p_{y} j + p_{z} k) \times$ $quad quad = (p_x i + p_y j + p_z k) xx$
$(q_{x} i + q_{y} j + q_{z} k)$ $quad quad quad quad (q_x i + q_y j + q_z k)$
Applying bilinear property of cross product
$= p_{x} i \times (q_{x} i + q_{y} j + q_{z} k)$ $quad quad = p_x i xx (q_x i + q_y j + q_z k)$
$+ p_{y} j \times (q_{x} i + q_{y} j + q_{z} k)$ $quad quad quad quad + p_y j xx (q_x i + q_y j + q_z k)$
$+ p_{z} k \times (q_{x} i + q_{y} j + q_{z} k)$ $quad quad quad quad + p_z k xx (q_x i + q_y j + q_z k)$
Applying $i \times i = 0$ $i xx i = 0$ , $j \times j = 0$ $j xx j = 0$ , $k \times k = 0$ $k xx k = 0$
$i \times j = k$ $i xx j = k$ , $j \times k = i$ $j xx k = i$ , $k \times i = j$ $k xx i = j$
$j \times i = - k$ $j xx i = -k$ , $k \times j = - i$ $k xx j = -i$ , $i \times k = - j$ $i xx k = -j$

After algebraic rearrangement, we get
$\vec{p} \times \vec{q}$ $vec p xx vec q$
$= (p_{y} q_{z} - p_{z} q_{y}) i$ $quad quad = (p_yq_z-p_zq_y)i$
$+ (p_{z} q_{x} - p_{x} q_{z}) j$ $quad quad quad quad + (p_zq_x-p_xq_z)j$
$+ (p_{x} q_{y} - p_{y} q_{x}) k$ $quad quad quad quad + (p_xq_y-p_yq_x)k$

summary

Vector Cross Product: for vectors $\vec{p}, \vec{q} \in ℝ^{3}$ $vec p, vec q in RR^3$
$\vec{p} \times \vec{q} = | \begin{matrix} i & j & k \\ p_{x} & p_{y} & p_{z} \\ q_{x} & q_{y} & q_{z} \end{matrix} |$ $vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z) |$

Vector Cross Product between two vectors in component form is defined as a vector from a determinant.

Outline

The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

• Introduction to Vectors

→ Introducing Vectors

→ Representation of Vectors

• Basic Properties of Vectors

→ Magnitude of Vectors

→ Types of Vectors

→ Properties of Magnitude

• Vectors & Coordinate Geometry

→ Vectors & Coordinate Geometry

→ Position Vector of a point

→ Directional Cosine

• Role of Direction in Vector Arithmetics

→ Vector Arithmetics

→ Understanding Direction of Vectors

• Vector Addition

→ Vector Additin : First Principles

→ Vector Addition : Component Form

→ Triangular Law

→ Parallelogram Law

• Multiplication of Vector by Scalar

→ Scalar Multiplication

→ Standard Unit Vectors

→ Vector as Sum of Vectors

→ Vector Component Form

• Vector Dot Product

→ Introduction to Vector Multiplication

→ Cause-Effect-Relation

→ Dot Product : First Principles

→ Dot Product : Projection Form

→ Dot Product : Component Form

→ Dot Product With Direction

• Vector Cross Product

→ Vector Multiplication : Cross Product

→ Cross Product : First Principles

→ Cross Product : Area of Parallelogram

→ Cross Product : Component Form

→ Cross Product : Direction Removed