Vector Scaling
The second main operation on vectors is scaling. Suppose k is any real number and u be any vector in or .
. Then scale the vector u by k and then drag the slider corresponding to k to see how the vector ku changes for different values of k.
- If k >0, then ku is the vector having the same direction as u such that its length is k times the length of u.
- If k = 0, then ku is a zero vector.
- If k < 0, then ku is the vector having the opposite direction to u such that its length is |k| times the length of u. (Note: |k| is the absolute value of k.)
. Then scale the vector u by k and then drag the slider corresponding to k to see how the vector ku changes for different values of k.
How is the column vector u related to the column vector ku ? Explain your answer briefly.
Vector Subtraction
Vector subtraction can be easily defined in terms of addition and scaling as follows: u - v = u + (-1)v. Also, the column vector u - v can be expressed in terms of the column vectors u and v using this definition.
You can construct vectors u and v in the above applet and then find out u + (-1)v.
Consider the parallelogram formed by two vectors u and v, can you express its two "diagonal vectors" in terms of u and v?