Vectors are mathematical entities that possess both magnitude and direction, crucial for representing quantities in various fields of science and engineering. In this section, we delve into the composition of vectors and explore the concept of collinearity through scalar multiplication.

Composition of Vectors

The composition of vectors involves adding two or more vectors together to form a single vector. This resultant vector encapsulates the combined effect of the original vectors.

Image courtesy of Ankplanet

Adding Vectors

To add vectors, align their components and combine them. For two vectors $a = (x_1, y_1)$ and $b = (x_2, y_2)$, the resultant vector $c = a + b$ is computed as:

Example 1:

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

This shows $c = (4, -2)$ as the sum of $a$ and $b$.

Graphical Representation

Graphically, to perform vector addition, place the tail of vector $b$ at the head of $a$. The vector $c$ spans from the tail of $a$ to the head of $b$.

Collinearity and Scalar Multiplication

Collinearity indicates that three or more points lie on a single straight line. Vectors express this through scalar multiplication, with vector $d$ being collinear with $a$ if $d = ka$, where $k$ is a scalar.

Image courtesy of Online Math Learning

Scalar Multiplication

Multiplying a vector by a scalar adjusts its magnitude, not its direction, unless the scalar is negative, which reverses the direction.

Example 2:

Given$a = (2, 3)$, find $d$ such that $d$ is thrice $a$.

Here, $d = (6, 9)$ is three times the length of $a$, directed similarly.

Assessing Collinearity

Collinearity can be deduced if vectors between points are scalar multiples of each other.

Example 3:

For points $A$, $B$, and $C$ with position vectors $a = (1, 2)$, $b = (2, 4)$, and $c = (3, 6)$, determine collinearity.

First, find vectors $AB$ and $AC$:

Since $AC = 2AB$, points $A$, $B$, and $C$ are collinear.

Practice Questions

Question 1:

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

Question 2:

If vector $m = (5, 0)$ and $n = k(2, -4)$ are collinear, find $k$.

To find $k$, equate $m$ to $k \times n$:

Question 3:

Prove that points $X$, $Y$, and $Z$ with position vectors $x = (0, 0)$, $y = (3, 3)$, and $z = (6, 6)$ are collinear.

Comparing vectors $XY$ and $XZ$:

Since $XZ = 2 \times XY$, $X$, $Y$, and $Z$ are collinear with the scalar multiplication factor of $2$