📑Examples of Vectors

Let's look at some specific vectors and what they represent:

• Vector $a$:

• Move 1 unit to the right and 3 units up.
• Vector $b$:

• Move 5 units to the right and 2 units down.
• Vector $c$:

• Move 3 units to the left and 2 units down.
• Vector $d$:

• Move 0 units to the right (i.e., don't move horizontally) and 3 units up.

Visual Representation on a Grid

When vectors are drawn on a grid, they start from an initial point (often called the "tail") and point towards a terminal point (called the "head"). The direction and length of the arrow represent the movement described by the vector.

• Vector $a$ moves from its initial point 1 unit to the right and 3 units up.
• Vector $b$ moves 5 units to the right and 2 units down.
• Vector $c$ moves 3 units to the left and 2 units down.
• Vector $d$ moves straight up, with no horizontal movement.

Example: Calculating the Magnitude of Another Vector

Let's take another vector $b=\begin{pmatrix} 3 \\ 4 \end{pmatrix}$:

1. Identify the Components:

• The vector $b$ has components 3 (right movement) and 4 (upward movement).

2. Apply Pythagoras' Theorem:

• The magnitude $∣b∣$ is calculated as:

• Calculate each component:

• Add them together:

• Take the square root:

Visual Representation

Important Note:

• Negatives and Squaring: When squaring the components, negatives become positive. This is why you do not need to worry about the signs of the components when calculating the magnitude.

Example 1: Adding Vectors $a$ and $b$

Given:

To find the resultant vector $a+b$:

3. Add the Top Numbers:

4. Add the Bottom Numbers:

5. Resultant Vector:

• This means that moving 3 units right and 3 units up (vector a) combined with moving 4 units right and 1 unit up (vector $b$) results in a movement of 7 units right and 4 units up.

Example 2: Adding Vectors $c$ and $d$

Given:

To find the resultant vector $c+d$:

6. Add the Top Numbers:

7. Add the Bottom Numbers:

8. Resultant Vector:

• This means that moving 5 units left and 2 units up (vector $c$) combined with moving 2 units left and 4 units up (vector $d$) results in a movement of 7 units left and 6 units up.

Visual Representation

Important Note:

• Watch Out for Negatives: Always be careful with negative numbers when adding vectors. Adding a negative number is the same as subtracting the positive number.

example, if $a=\begin{pmatrix} 4 \\ -2 \end{pmatrix}$, then the negative of $a$ is $-a=\begin{pmatrix} -4 \\ 2 \end{pmatrix}$.

Example

Subtracting Vectors $p$ and $q$

Given:

To find $p−q$:

9. Find the Negative of $q$:

10. Add $p$ and $−q$:

• Add the corresponding components:

11. Resultant Vector:

• This resultant vector indicates a movement of 1 unit to the left and 2 units up.

Visual Representation

Important Note:

• Direction Reversal: When subtracting, the direction of the vector being subtracted is reversed.
• Adding the Negative: Subtraction of vectors can always be handled by adding the negative of the vector you wish to subtract.

Example 1: Multiplying Vector $p$ by $2$ Given:

$p=\begin{pmatrix} 2 \\ 2 \end{pmatrix}$

To find $2p$:

Multiply the Top Number:

Multiply the Bottom Number:

Resultant Vector:

• This means that the vector $p$ has been scaled up by 2, resulting in a movement of 4 units right and 4 units up.

Example 2: Multiplying Vector $q$ by $3$ Given:

$q=\begin{pmatrix} 3 \\ 2 \end{pmatrix}$

To find $3q$:

Multiply the Top Number:

Multiply the Bottom Number:

Resultant Vector:

• The vector $q$ is now scaled by 3, giving a movement of 9 units right and 6 units up.

Example 3: Multiplying Vector $r$ by $-4$ Given:

$r=\begin{pmatrix} 1 \\ 2 \end{pmatrix}$

To find $−4r$:

Multiply the Top Number:

Multiply the Bottom Number:

Resultant Vector:

• Here, the vector $r$ has been scaled by -4, reversing its direction and resulting in a movement of 4 units left and 8 units down.

Visual Representation

Important Note:

Scaling and Direction: Positive scalars stretch the vector in the same direction, while negative scalars reverse the direction.

Consistency: Always apply the scalar to both components (top and bottom) of the vector.

📑Example: Given Vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -4 \\ 2 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} -1 \\ -2 \end{pmatrix}$

Problem (a): Calculate $4a+3b+c$

12. Multiply each vector by the scalar:

13. Add the vectors:

• Add the corresponding components:

14. Resultant Vector:

Problem (b): Calculate $2a−5b−2c$

15. Multiply each vector by the scalar:

16. Subtract the vectors:

• Subtract the corresponding components:

17. Resultant Vector:

Important Note:

• Care with Signs: Be especially careful with negative signs when multiplying and adding/subtracting vectors.
• Scalar Multiplication: Always multiply each component of the vector by the scalar before adding or subtracting.

📑Example: Describing Routes in a Regular Hexagon

Consider the regular hexagon $ABCDEF$ with vectors $a$ and $b$ as shown:

• Vector $a$: Represents the vector from $A$ to $B$.
• Vector $b$: Represents the vector from $B$ to $C$.

We'll describe the routes $FC$, $DA$, and $EB$ in terms of these vectors.

Example

(i) Describing $\overrightarrow{FC}$

• Route: The best way to go from $F$ to $C$ is straight across the middle, because we know each horizontal line in the hexagon is the same length.
• Calculation:

(ii) Describing $\overrightarrow{DA}$

• Route: It might be tempting to go directly from $D$ to $A$, but that route isn't known. Instead, we'll need to go the opposite way along the given vector $a$.
• Calculation:

(iii) Describing $\overrightarrow{EB}$

• Route: Directly across the middle looks good, but again, that route isn't defined by known vectors. Instead, we must take the longer route:

• First, go from $E$ to $F$ using vector $b$.

• Then from $F$ to $A$ using vector $−a$.

• Finally, from $A$ to $B$ using vector $a$.

• Calculation: