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Constructions: N/A
Generalizations: 4.2 Column Vector Addition

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Shifting a Vertex Model

Recall our triangle from 3.1.1 Vertex Model of Points. We can represent the three corners of our triangle with column vectors.

\begin{aligned} {\vec{v}}_{1} = [\begin{array}{c} 0 \\ 0 \end{array}] & {\vec{v}}_{2} = [\begin{array}{c} - 1 \\ 1 \end{array}] & {\vec{v}}_{3} = [\begin{array}{c} 2 \\ 1 \end{array}] \end{aligned}

Create a second triangle by shifting the vertices to the left by one unit and down by two units using 4.2 Column Vector Addition.

Solution. Let's create the vector $\vec{s}$ to shift the triangle.

\vec{s} = [\begin{matrix} - 1 \\ - 2 \end{matrix}]

Now, we can shift our vertices. If we let ${\vec{x}}_{i} = {\vec{v}}_{i} + \vec{s}$ , we see:

\begin{aligned} {\vec{x}}_{1} = [\begin{array}{c} - 1 \\ - 2 \end{array}] & {\vec{x}}_{2} = [\begin{array}{c} - 2 \\ - 1 \end{array}] & {\vec{x}}_{3} = [\begin{array}{c} 1 \\ - 1 \end{array}] \end{aligned}

If we graph these points, we indeed see that the vertices were shifted left one and down two.

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