10th Class Math Chapter 6 Notes PDF 2026 – Vectors in Plane

Definition

A vector $A B$ has a tail ( $A$ ) and a tip ( $B$ ), giving it both magnitude and direction.

In component form, $A B = (x_{2} - x_{1}) i^+ (y_{2} - y_{1}) j^$ , where $i^, j^$ are the unit vectors along the $x$ - and $y$ -axes.

Example 1 — writing a vector in component form

Write $A B$ for $A (1, - 7)$ , $B (- 2, 4)$ .

$x$ -component: $- 2 - 1 = - 3$ .
$y$ -component: $4 - (- 7) = 11$ .

Answer: $A B = - 3 i^+ 11 j^$

$∣ a ∣ = x^{2} + y^{2}$ for $a = x i^+ y j^$

Unit vector in the direction of $a$ : $a^= \frac{a}{∣ a ∣}$

Example 2 — magnitude and unit vector

Find the magnitude and unit vector for $a = 6 i^+ 8 j^$ .

$∣ a ∣ = 6^{2} + 8^{2} = 100 = 10$ .
$a^= \frac{6 i ^ + 8 j ^}{10}$ .

Answer: $∣ a ∣ = 10$ ; $a^= \frac{3}{5} i^+ \frac{4}{5} j^$

Rule

Addition/subtraction: combine $i^$ -parts together and $j^$ -parts together.

Scalar multiplication $k a$ : multiply every component by $k$ — scales the length, and reverses direction if $k < 0$ .

Two vectors are parallel exactly when one is a scalar multiple of the other.

Example 3 — checking for parallel vectors

Are $a = 6 i^+ j^$ and $b = 12 i^+ 2 j^$ parallel?

Check if $b = k a$ : $12 = 6 k \Rightarrow k = 2$ ; check the other component: $2 = 1 (2)$ — matches.

Answer: Yes, $b = 2 a$ — parallel

Example 4 — combining vectors

If $a = 7 i^- 3 j^$ and $b = i^+ 5 j^$ , find $2 a - b$ .

$2 a = 14 i^- 6 j^$ .
$2 a - b = (14 - 1) i^+ (- 6 - 5) j^$ .

Answer: $13 i^- 11 j^$

Since $A B = D C$ exactly when $A B$ and $D C$ are parallel and equal in length, vectors give a clean way to prove a quadrilateral is a parallelogram, or that two sides of a triangle are equal (isosceles).

Example 5 — proving a parallelogram

Outline how to show $P QR S$ is a parallelogram using vectors.

Find $P Q$ and $S R$ in component form.
If $P Q = S R$ , then $P Q$ and $S R$ are equal and parallel — so $P QR S$ is a parallelogram.

Answer: Compute both vectors and compare

Velocity, force and displacement are all vector quantities in real applications — combining them (e.g. a plane's airspeed plus a crosswind) is ordinary vector addition.

Example 6 — combining velocities

A plane has airspeed $v_{p} = 20 i^$ and faces a crosswind $v_{w} = 5 j^$ . Find its resultant velocity.

Add component-wise: $v = v_{p} + v_{w} = 20 i^+ 5 j^$ .

Answer: $v = 20 i^+ 5 j^$ (magnitude $425 \approx 20.6$ )

Quick Recap

● $A B = (x_{2} - x_{1}) i^+ (y_{2} - y_{1}) j^$
● $∣ a ∣ = x^{2} + y^{2}$ ; unit vector $a^= a /∣ a ∣$
●Add/subtract component-wise; $k a$ scales length and flips direction if $k < 0$
●Parallel vectors: one is a scalar multiple of the other
● $A B = D C$ proves $A B C D$ -type equal, parallel sides — the vector route to parallelogram/isosceles proofs
●Real vector quantities (velocity, force) combine by ordinary vector addition

10th Class Math Chapter 6 Notes

Vector Notation

Magnitude and Unit Vectors

Vector Operations

Vectors and Coordinate Geometry

Applications

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