Vector Algebra — Part 1: Vectors, components and the section formula: visual lesson, ISC Class 12 Mathematics

Lesson

Vector Algebra, Part 1

Quantities with both a size and a direction.

In this lesson you will

Find the magnitude, unit vector, direction cosines and direction ratios of a vector
Recognise zero, unit, equal, negative and collinear vectors
Add, subtract and scale vectors in component form
Write the vector joining two points from their position vectors
Find the point dividing a segment internally or externally in a given ratio

Part 1 at a glance.

Concept

What you will learn

Magnitude, unit vectors and direction cosines
Types of vectors
Components and position vectors
The section formula, internal and external

$\vec{v} = \overrightarrow{PQ}$: the length of the arrow is $|\vec{v}|$; the arrowhead gives the direction.

Concept

Magnitude and direction cosines

For $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ : $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$ , and the direction cosines are the components of the unit vector $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$ .

l^2 + m^2 + n^2 = 1

Ratios are proportional to the cosines; cosines are fixed.
Quantity	Formula	Note
Magnitude	$\|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2}$	always $\ge 0$
Direction cosines	$l = \dfrac{a_1}{\|\vec{a}\|},\ m = \dfrac{a_2}{\|\vec{a}\|},\ n = \dfrac{a_3}{\|\vec{a}\|}$	$l^2 + m^2 + n^2 = 1$
Direction ratios	$a_1, a_2, a_3$ (or any multiple)	not unique

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