Vectors play a crucial role in both physics and mathematics, offering a way to represent quantities that have both magnitude and direction. In this section, we will explore the fundamental operations involving vectors: vector addition and scalar multiplication. Understanding these operations is key to mastering the study of vectors.

Vector Addition

Vector addition combines two vectors to form a single vector. This operation follows the principle of the triangle or parallelogram law of addition.

• Definition: For vectors $a = (x_1, y_1)$ and $b = (x_2, y_2)$, the sum $a + b$ is given by $(x_1+x_2, y_1+y_2)$.

• Graphical Representation: Vectors can be added graphically by placing the tail of vector $b$ at the head of vector $a$. The resultant vector $a + b$ is then drawn from the tail of $a$ to the head of $b$.

Image courtesy of Cuemath

Example: Addition of Vectors $a$ and $b$

Given vectors $a = (3, 2)$ and $b = (1, 4)$, calculate $a + b$.

$a + b = (3 + 1, 2 + 4) = (4, 6)$

This result means that by adding vector $a$ to vector $b$, we get a new vector $a + b$ with an x-component of 4 and a y-component of 6.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), affecting the magnitude of the vector without changing its direction (unless the scalar is negative, which reverses the direction).

• Definition: For a vector $a = (x_1, y_1)$ and a scalar $c$, the product $c \cdot a$ is given by $(c \cdot x_1, c \cdot y_1)$.

• Physical Interpretation: Scalar multiplication can be seen as scaling the length of the vector by the scalar $c$.

Image courtesy of Wikipedia

Example: Multiplication of Vector (a) by Scalar (c)

Given vector $a = (3, 2)$ and scalar $c = 2$, find $c \cdot a$.

$c \cdot a = 2 \cdot (3, 2) = (6, 4)$

This calculation demonstrates that multiplying vector $a$ by the scalar $c = 2$ results in a new vector whose magnitude is doubled, resulting in the vector $(6, 4).$

Practice Problems

Problem 1: Addition of Vectors $p$ and $q$

Given vectors $p = (5, -3)$and $q = (-2, 4)$, calculate $p + q$.

Solution:

$p + q = (5 - 2, -3 + 4) = (3, 1)$

Problem 2: Multiplication of Vector $r$ by Scalar $d$

Given vector $r = (4, -5)$and scalar $d = -3$, find $d \cdot r$.

Solution:

$d \cdot r = -3 \cdot (4, -5) = (-12, 15)$