The second main operation on vectors is [b]scaling[/b]. Suppose k is any real number and u be any vector in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math]. [br][list][*]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.[br][/*][*]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.)[/*][/list][br]In the applet below, you can construct a vector u in [math]\mathbb{R}^2[/math] using the vector tool [icon]/images/ggb/toolbar/mode_vector.png[/icon]. 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.[br]
How is the column vector u related to the column vector ku ? Explain your answer briefly.
The column vector ku can be obtained by multiplying k to all entries of column vector u i.e. [math]k\begin{pmatrix}u_x\\u_y\end{pmatrix}=\begin{pmatrix}ku_x\\ku_y\end{pmatrix}[/math].
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.[br][br]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?
The two diagonal vectors are u + v and u - v. And u - v is the diagonal vector pointing from the arrowhead of v to the arrowhead of u.