Vector Algebra : Cross Product : Area of Parallelogram

maths > vector-algebra

Cross Product : Area of Parallelogram

what you'll learn...

Overview

Vector Cross Product : Area of Parallelogram Form

» component in perpendicular is calculated by the angle
    → $\vec{p} \times \vec{q}$ $vec p xx vec q$ $= | \vec{p} | | \vec{q} | \sin θ \hat{n}$ $= |vec p| |vec q| sin theta hat n$
    → Note that $| \vec{b} | = | \vec{q} | \sin θ$ $|vec b| = |vec q| sin theta$
    → $| \vec{p} | | \vec{b} |$ $|vec p||vec b|$ is the area of parallelogram with sides $\vec{p}$ $vec p$ and $\vec{q}$ $vec q$

parallelogram

Vector cross product is understood as product between components in perpendicular. In the figure, The magnitude of $\vec{p} \times \vec{q}$ $vec p xx vec q$ is 'the area of the parallelogram OLMN'.

Area of a parallelogram = base $\times$ $xx$ height
The base is $| p |$ $|p|$ and the height is $| b |$ $|b|$ .

Two vectors $\vec{p}$ $vec p$ and $\vec{q}$ $vec q$ with magnitudes $2$ $2$ and $3$ $3$ respectively are at an angle $30^{\circ}$ $30^@$ . What is the area of the parallelogram made by $\vec{p}$ $vec p$ and $\vec{q}$ $vec q$ ?

The answers are
$2 \times 3 \times {\sin 30}^{\circ}$ $2 xx 3 xx sin 30^@$
$2 \times 3 \times \frac{1}{2}$ $2 xx 3 xx 1/2$
$= 3$ $=3$

summary

Area of a Parallelogram made by two vectors: Given $\vec{p}$ $vec p$ and $\vec{q}$ $vec q$ , the area of the parallelogram made by them is
$= | \vec{p} \times \vec{q} |$ $= |vec p xx vec q|$ Vector cross product can be used to find area of the Parallelogram made by two vectors.

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