Introduction

• Vectors are quantities with both direction and length.
• Adding vectors tells us the combined effect of two or more movements, forces, or changes.
• This article shows two friendly ways to add vectors:
  • Geometrically (drawing pictures)
  • Algebraically (using numbers)

What a Vector Represents in Simple Terms

• A vector can represent:
  • A movement (e.g., “walk 3 steps east”)
  • A force (e.g., “push 5 N upward”)
  • A velocity (e.g., “10 m/s north”)
• Think of a vector as an arrow:
  • Length = how big the quantity is
  • Direction = which way it points

Why Adding Vectors Matters

• Many real‑world situations involve combining effects:
  • Two forces acting on an object
  • Two movements happening one after another
  • Wind + airplane velocity
• Vector addition gives the overall result.

Geometric Addition: The Tip‑to‑Tail Method

• This is the most visual way to add vectors.
• Steps:
  • Draw the first vector as an arrow.
  • Place the tail of the second vector at the tip of the first.
  • Draw the resultant vector from the tail of the first to the tip of the second.
• The final arrow is the sum: $$\vec{a} + \vec{b} = \vec{r}$$

Understanding Vector Components

• Any vector in a plane can be broken into horizontal and vertical parts:
  • If a vector has length $L$ and angle $\theta$:
    • Horizontal component: $L\cos\theta$
    • Vertical component: $L\sin\theta$
• Components let us work with vectors using simple arithmetic.

Algebraic Addition Using Components

• If $$\vec{a} = (a_x, a_y), \quad \vec{b} = (b_x, b_y),$$ then $$\vec{a} + \vec{b} = (a_x + b_x,\; a_y + b_y).$$
• Steps:
  • Add the $x$‑components.
  • Add the $y$‑components.
  • The result is the new vector.

Common Mistakes and How to Avoid Them

• Mistake: Adding lengths but ignoring direction.
  Fix: Always consider direction; vectors are not just numbers.
• Mistake: Drawing arrows without consistent scale.
  Fix: Use the same scale for all vectors in a diagram.
• Mistake: Mixing up components.
  Fix: Keep $x$ and $y$ components separate and labelled.

Visual Intuition: How Direction and Length Combine

• Adding vectors is like combining movements:
  • Move along $\vec{a}$, then along $\vec{b}$ → total movement is $\vec{a} + \vec{b}$.
• If vectors point in similar directions:
  • The sum is longer.
• If they point in opposite directions:
  • The sum is shorter or may even reverse direction.
• If they are perpendicular:
  • The sum forms the hypotenuse of a right triangle.

Practice Problems Worked Step‑by‑Step

• Example 1:
  $\vec{a} = (3, 1)$ and $\vec{b} = (2, 4)$
  • Add components:
    • $x$: $3 + 2 = 5$
    • $y$: $1 + 4 = 5$
  • Result: $\vec{a} + \vec{b} = (5, 5)$
• Example 2:
  Two vectors of lengths 4 and 3 at right angles.
  • Resultant length: $$\sqrt{4^2 + 3^2} = 5.$$

Calculator

Exercises

Try these on your own. Draw diagrams where helpful.

1. Add the vectors $\vec{a} = (1, 4)$ and $\vec{b} = (3, -2)$.

   Solution

   Add the vectors \(\vec{a} = (1, 4)\) and \(\vec{b} = (3, -2)\)

   • \(x\)-components: \(1 + 3 = 4\)
   • \(y\)-components: \(4 + (-2) = 2\)
   • Solution: \((4, 2)\)
2. Two vectors of lengths 6 and 8 point in the same direction. What is their sum’s length?

   Solution

   Two vectors of lengths 6 and 8 point in the same direction. What is the sum’s length?

   • Same direction → lengths simply add.
   • Solution: \(6 + 8 = 14\)
3. Two vectors of lengths 5 and 7 point in opposite directions. What is the length of the sum?

   Solution

   Two vectors of lengths 5 and 7 point in opposite directions. What is the length of the sum?

   • Opposite directions → subtract lengths.
   • Solution: \(|7 - 5| = 2\)
4. Add the vectors $\vec{u} = (-2, 3)$ and $\vec{v} = (4, 1)$.

   Solution

   Add the vectors \(\vec{u} = (-2, 3)\) and \(\vec{v} = (4, 1)\)

   • \(x\)-components: \(-2 + 4 = 2\)
   • \(y\)-components: \(3 + 1 = 4\)
   • Solution: \((2, 4)\)
5. A vector of length 10 makes a $30^\circ$ angle with the horizontal. Find its components.

   Solution

   A vector of length 10 makes a \(30^\circ\) angle with the horizontal. Find its components.

   • Horizontal: \(10\cos 30^\circ = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}\)
   • Vertical: \(10\sin 30^\circ = 10 \cdot \frac{1}{2} = 5\)
   • Solution: \((5\sqrt{3},\, 5)\)
6. Draw two perpendicular vectors of lengths 3 and 4. Use geometry to find the resultant.

   Solution

   Draw two perpendicular vectors of lengths 3 and 4. Use geometry to find the resultant.

   • Right triangle with legs 3 and 4.
   • Resultant length:
     \[
     \sqrt{3^2 + 4^2} = 5
     \]
   • Solution: Resultant length is 5.
7. Add $\vec{p} = (0, 5)$ and $\vec{q} = (-3, -1)$.

   Solution

   Add \(\vec{p} = (0, 5)\) and \(\vec{q} = (-3, -1)\)

   • \(x\)-components: \(0 + (-3) = -3\)
   • \(y\)-components: \(5 + (-1) = 4\)
   • Solution: \((-3, 4)\)
8. A person walks 4 km east, then 2 km north. Draw and label the resultant vector.

   Solution

   A person walks 4 km east, then 2 km north. Draw and label the resultant vector.

   • Components: \((4, 2)\)
   • Length:
     \[
     \sqrt{4^2 + 2^2} = \sqrt{20} = 2\sqrt{5}
     \]
   • Solution: Resultant vector is \((4, 2)\) with length \(2\sqrt{5}\).
9. Add the vectors $\vec{a} = (7, -3)$ and $\vec{b} = (-2, 5)$.

   Solution

   Add the vectors \(\vec{a} = (7, -3)\) and \(\vec{b} = (-2, 5)\)

   • \(x\)-components: \(7 + (-2) = 5\)
   • \(y\)-components: \(-3 + 5 = 2\)
   • Solution: \((5, 2)\)
10. Two vectors form a $60^\circ$ angle and have lengths 5 and 5. estimate the resultant length.

    Solution

    Two vectors form a \(60^\circ\) angle and have lengths 5 and 5. Estimate the resultant length.

    Using the law of cosines:
    \[
    R = \sqrt{5^2 + 5^2 + 2\cdot 5\cdot 5\cos 60^\circ}
    \]
    \[
    R = \sqrt{25 + 25 + 50 \cdot \tfrac{1}{2}} = \sqrt{25 + 25 + 25} = \sqrt{75}
    \]
    \[
    R = 5\sqrt{3} \approx 8.66
    \]

    • Solution: Approximately 8.66.