Question

Use a rectangular coordinate system to plot u, v, and their images under the given transformation t. Describe geometrically what t does to each vector x in?

Answer 1

The transformation ( t ) in the rectangular coordinate system alters vectors ( u ) and ( v ) in a specific way. It rotates ( u ) counterclockwise by
`\( 90^\circ \)` and stretches ( v ) by a factor of ( 2 ), keeping its direction unchanged.

Step-by-step explanation:

In a rectangular coordinate system, let ( u = (2, 3) ) and ( v = (-1, 4) ) be the vectors. The transformation ( t ) can be described as follows:

( t(u) ) rotates vector ( u ) counterclockwise by
`\( 90^\circ \)`. To compute the image of ( u ), ( t(u) ), we can use the rotation matrix for a
`\( 90^\circ \)` counterclockwise rotation in the ( xy )-plane:

`\[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} -3 \\ 2 \end{bmatrix} \]`

So, ( t(u) = (-3, 2) ). Geometrically, ( t ) rotates vector ( u ) counterclockwise by
`\( 90^\circ \)`.

Next, ( t(v) ) stretches vector ( v ) by a factor of ( 2 ) while keeping its direction unchanged. Thus,

[ t(v) = 2 \cdot (-1, 4) = (-2, 8) ]

Hence, ( t(v) = (-2, 8) ). This means that ( t ) stretches vector ( v ) by a factor of ( 2 ) while maintaining its original direction.

In summary, the transformation ( t ) rotates vector ( u ) counterclockwise by
`\( 90^\circ \)`and stretches vector ( v ) by a factor of ( 2 ) without altering its direction in the rectangular coordinate system.