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Introduction to two-dimensional motion: vector review

Review the key concepts and skills for two-dimensional motion, including how to add and subtract vectors.

Key terms

ResultantSum of two or more vectors
Vector componentsHorizontal and vertical parts of a vector

How to add and subtract vectors

Vector addition

To add vectors we can use the head to tail method (Figure 1).
  1. Place the tail of one vector at the tip of the other vector.
  2. Draw an arrow from the tail of the first vector to the tip of the second vector. This new vector is the sum of the first two vectors.
Figure 1. Adding vectors start text, A, end text and start text, B, end text using the head to tail method

Vector subtraction

Subtraction of a vector is the same as adding the vector with the opposite direction (see Figure 2).
Figure 2. Subtracting vector start text, B, end text from start text, A, end text using the head to tail method to add start text, A, end text and minus, start text, B, end text
In other words, vector start text, negative, B, end text is the negative of vector start text, B, end text if they have the same length, but point in opposite directions. Using this trick, the head to tail method of adding vectors works for subtraction as well.

Finding perpendicular vector components

A vector can be represented as the sum of two perpendicular vector components. To find the horizontal and vertical components, we draw a right triangle with the vector as the hypotenuse D (see Figure 3). One way to choose the sides of the triangle is the following:
Figure 3. Finding horizontal and vertical vector components
  • The horizontal component (D, start subscript, x, end subscript) starts at the tail of D and is parallel to the horizontal axis. The head of the horizontal component has the same horizontal position as start text, D, end text.
  • The vertical component (D, start subscript, y, end subscript) starts at the head of D, start subscript, x, end subscript and ends at the head of the original vector. D, start subscript, y, end subscript is parallel to the vertical axis.

Common mistakes and misconceptions

  1. A common mistake is attempting to add vectors head to head, or tail to tail. However, this approach is incorrect, and gives a different answer from adding the vectors head to tail (Figure 4).
Figure 4. Combining vectors head to tail does not give the same answer as combining vectors head to head. Only head to tail is correct for adding vectors.
  1. Some people think vectors have to be added in a particular order. Since vector math is commutative, the order does not matter.

Learn more

To check your understanding and work toward mastering these concepts, check out the exercise on describing two-dimensional motion with vectors.

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