represent a vector as two perpendicular components

Scalars and Vectors

What is a scalar?

Scalars are quantities that only have magnitude. They don’t point in any direction. Examples: speed (50 km h⁻¹), mass (10 kg), temperature (30 °C).

What is a vector?

Vectors have both magnitude and direction. They are usually shown as arrows. Example: a velocity of 30 m s⁻¹ to the east. The arrow points east and its length represents 30 m s⁻¹.

Representing a vector with perpendicular components

Any vector can be split into two perpendicular components along the x‑axis and y‑axis. Think of a skateboarder moving diagonally across a field – the horizontal and vertical parts of his motion are the components.

Mathematically we write:

\$\vec{v} = vx\,\hat{i} + vy\,\hat{j}\$

where \$vx\$ is the horizontal component and \$vy\$ is the vertical component.

ComponentSymbolUnit
Horizontal\$v_x\$m s⁻¹
Vertical\$v_y\$m s⁻¹

Example: Finding components of a 10 m s⁻¹ velocity at 30° above the horizontal

  1. Identify the magnitude \$v = 10\$ m s⁻¹ and the angle \$\theta = 30^\circ\$.

  2. Compute the horizontal component: \$v_x = v \cos\theta\$.

  3. Compute the vertical component: \$v_y = v \sin\theta\$.

  4. Write the vector in component form.

Calculations:

\$v_x = 10 \cos 30^\circ = 8.66 \text{ m s}^{-1}\$

\$v_y = 10 \sin 30^\circ = 5.00 \text{ m s}^{-1}\$

Thus the vector is:

\$\vec{v} = 8.66\,\hat{i} + 5.00\,\hat{j}\ \text{m s}^{-1}\$

Exam Tip 🚀

When the exam asks for components:

  • Use \$vx = v \cos\theta\$ and \$vy = v \sin\theta\$ if the angle is measured from the x‑axis.

  • Check the quadrant of the angle to decide the sign of each component.

  • If components are given, you can find the magnitude with \$v = \sqrt{vx^2 + vy^2}\$ and the direction with \$\theta = \tan^{-1}(vy/vx)\$.

  • Always round to the correct number of significant figures as specified in the question.

Common Mistakes to Avoid

  • Mixing up \$\cos\$ and \$\sin\$ for the wrong component.

  • Forgetting to convert degrees to radians if the calculator is set to radian mode.

  • Using the wrong sign for components in the second or third quadrants.

  • Not checking significant figures or units in the final answer.

Remember: vectors are like arrows on a map – the length tells you how big, and the direction tells you where. Splitting them into perpendicular components is just like breaking a pizza slice into a horizontal and a vertical slice – each part is easier to handle, and together they give you the whole picture. 🍕📐