Multiplication by Scalar and Translation by Vector

Multiplication by Scalar and Translation by Vector

Example: In the figure, direction of $\vec{p}$

​ is 3 square units to the right and 4 square units up. Find the relation of other vectors with $\vec{p}$

​.

Solution: (i) $\vec{q} = 2\vec{p}$

​=2p

​ (ii) $\vec{r} = -1.5\vec{p}$

=−1.5p

​ (iii) $\vec{s} = -0.5\vec{p}$

=−0.5p

​ (iv) $\vec{t} = 0.5\vec{p}$

=0.5p

​ (v) $\vec{u} = -\vec{p}$

=−p

​

Translation by Vector

Translation by a vector is a process that moves a point or figure in space by a given vector. It is a type of transformation that slides a figure without rotating it, so the shape, size, and orientation of the figure do not change.

Geometrically: Place the tail of the arrow at the point, and the translated point is at the head of the arrow.

Algebraically: Represent the points and the vector by their coordinates. The translation vector's coordinates indicate how far to move the points along each coordinate axis.

For example, the vector [a,b] moves points a units horizontally and b units vertically.

For example, a translation vector that moves a figure 4 units right and 2 units down can be represented mathematically as [4, -2].

Example:

Use the translation vector [2, -1] to find the new coordinates of triangle ABC when A = (-3,2), B = (-1,0), and C = (-2,-1).

Solution: The vector indicates a translation of 2 units to the right and 1 unit down. We can therefore add 2 to all of the x-values and subtract 1 from all of the y-values to find the vertices of the image.

A' = (-3+2, 2-1) = (-1,1)

B' = (-1+2, 0-1) = (1,-1)

C' = (-2+2, -1-1) = (0,-2)

Parallel Vectors