Adding and Subtracting Vectors

To add or subtract two vectors, add or subtract the corresponding components.

Let be two vectors.

Then, the sum of is the vector

The difference of is

The sum of two or more vectors is called the resultant. The resultant of two vectors can be found using either the parallelogram method or the triangle method.

Parallelogram Method:

Draw the vectors so that their initial points coincide. Then draw lines to form a complete parallelogram. The diagonal from the initial point to the opposite vertex of the parallelogram is the resultant.

Vector Addition:

  1. Place both vectors at the same initial point.

  2. Complete the parallelogram. The resultant vector is the diagonal of the parallelogram.

Vector Subtraction:

  1. Complete the parallelogram.

  2. Draw the diagonals of the parallelogram from the initial point.

Triangle Method:

Draw the vectors one after another, placing the initial point of each successive vector at the terminal point of the previous vector. Then draw the resultant from the initial point of the first vector to the terminal point of the last vector. This method is also called the head-to-tail method .

Vector Addition:

  1. Place the initial point of the second vector at the terminal point of the first vector .

  2. The resultant connects the initial point of the first vector and the terminal point of the second vector .

Vector Subtraction:

  1. Place the initial point of at the terminal point of .

  2. Draw the vector from the initial point of to the terminal point of .

Example:

By the definition of vector addition, Substitute the given values of u1 , u2 , v1 and v2 into the definition of vector addition.

We know that the difference u v is equivalent to the sum u + (– v ). We will need to determine the components of – v .

Recall that – v is a scalar multiple of –1 times v . From the definition of scalar multiplication, we know:

Substituting the given values of the components of v , we obtain

Substitute the values of the components of u and – v into the definition of vector addition.

We substitute the values of the components of u and – v into the definition of vector addition and simplify: