← Back to CoursesNewtonian Mechanics: Beginner

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Vectors and Components

Manual: General · Subject: Newtonian Mechanics

Introduce vectors so motion and forces can be described in two and three dimensions.

Why Scalars Are Not Enough

What is a vector?

A vector is a quantity with magnitude and direction. Displacement, velocity, acceleration, and force are vectors.

Components

A vector can be split into parts along chosen axes. If a force F⃗\vec{F} makes an angle θ\theta with the horizontal, then Fx=Fcos⁡θF_x = F\cos\theta and Fy=Fsin⁡θF_y = F\sin\theta.

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Why components help

Breaking one angled vector into horizontal and vertical parts turns a 2D problem into two simpler 1D problems.

Scalar vs. vector

Scalar

  • Magnitude only
  • Examples: mass, time, speed

Vector

  • Magnitude and direction
  • Examples: displacement, velocity, force

Adding vectors

Vectors add according to direction, not just size. In components, you add xx parts together and yy parts together separately.

Which of the following is a vector?

If a vector has magnitude FF and angle θ\theta from the horizontal, what is its horizontal component?

Component method

  1. 1

    Step 1: Draw the vector on coordinate axes.

  2. 2

    Step 2: Identify the angle relative to the axes.

  3. 3

    Step 3: Use sine and cosine to find components.

  4. 4

    Step 4: Add components separately when combining vectors.