Addition, subtraction and scalar multiplication of vectors is very straight forward. Vector multiplication and vector division, however, do not make sense and therefore do not have a definition.


Vector Addition

Parallelogram Law

To add two vectors together, one simply adds together the corresponding components. For example:

\left(\begin{array}{c}2\\-5\end{array}\right)+\left(\begin{array}{c}4\\3\end{array}\right)=\left(\begin{array}{c}6\\-2\end{array}\right)

The vector that results from applying one vector followed by another by adding, i.e. a+b, is the vector that points directly from the start point to the finish point. Applying the vectors the other way round, i.e. b+a, also results in the same resultant vector. This is known as the parallelogram law of vector addition. See figure.

This may also be written as \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC} if considering vectors between points A, B and C.


Vector Subtraction

Triangle Law

As you might expect, vector subtraction is achieved by subtracting the corresponding components of the given vectors. For example:

\left(\begin{array}{c}3\\-5\end{array}\right)-\left(\begin{array}{c}-4\\2\end{array}\right)=\left(\begin{array}{c}7\\-7\end{array}\right)

Rather than starting from a point and moving along two vectors one after the other, consider two vectors starting from the same point, say a and b. The vector that results from subtracting a from b is the one that points directly from the end of a to the end of b. See figure. This is known as the triangle law of vector addition. It can be thought of as starting at the end of a, moving backwards along a and then forwards along b (-a+b but written as b-a). See Position Vectors for more on this.


Scalar Multiplication of Vectors

Parallel Vectors

There is no definition for multiplying vectors together but we can multiply or divide vectors by a scalar (ie a single number). For example,

2\left(\begin{array}{c}-1\\6\end{array}\right)=\left(\begin{array}{c}-2\\12\end{array}\right)  or  -\frac{1}{3}\left(\begin{array}{c}3\\-2\end{array}\right)=\left(\begin{array}{c}-1\\\frac{2}{3}\end{array}\right)

As we can see from the diagram, scalar multiples of vectors are all parallel.

Since scalar multiplication and vector addition is possible, it follows that any vector can be expressed as a linear combination of the standard unit vectors. For example,

\left(\begin{array}{c}7\\-5\end{array}\right)=\left(\begin{array}{c}7\\0\end{array}\right)-\left(\begin{array}{c}0\\5\end{array}\right)=7\left(\begin{array}{c}1\\0\end{array}\right)-5\left(\begin{array}{c}0\\1\end{array}\right)=7{\bf i}-5{\bf j}

This is useful when writing vectors on a single line rather than stacked horizontally.


Example

Consider the following vectors:

{\bf a}=\left(\begin{array}{c}-1\\3\end{array}\right),     {\bf b}=\left(\begin{array}{c}8\\5\end{array}\right),    {\bf c}=\left(\begin{array}{c}4\\-2\end{array}\right).

The vector 6{\bf a}+4{\bf b}-9{\bf c} is parallel to the vector 5{\bf i}+k{\bf j}. Find the value of k.

See more on problem solving with vectors.


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