Questions: Give the component form of a vector that maps ΔJKL to ΔJ'K'L', J(-5,2), K(-1,-3), L(1,5) J'(-3,-1), K'(1,-6), L'(3,2) J(3,6), K(4,-1), L, , 2,2 J'(6,7), K'(7,0), L'(3,3)

Transcript text: Give the component form of a vector that maps $\Delta J K L$ to $\Delta J^{\prime} K^{\prime} L^{\prime}$, $J(-5,2), K(-1,-3), L(1,5)$ $J^{\prime}(-3,-1), K^{\prime}(1,-6), L^{\prime}(3,2)$ $J(3,6), K(4,-1), L, \quad, \quad, 2,2$ $J^{\prime}(6,7), K^{\prime}(7,0), L^{\prime}(3,3)$

Solution

Solution Steps

To find the component form of a vector that maps $\Delta J K L$ to $\Delta J^{\prime} K^{\prime} L^{\prime}$, we need to determine the translation vector that moves each point of the original triangle to the corresponding point in the transformed triangle. This can be done by subtracting the coordinates of each point in the original triangle from the coordinates of the corresponding point in the transformed triangle.

Solution Approach

Calculate the translation vector for each pair of corresponding points.
Verify that the translation vectors are consistent for all points.

Step 1: Identify the Initial and Final Coordinates

We are given the initial coordinates of the vertices of triangle $ \Delta JKL $ and the final coordinates of the vertices of triangle $ \Delta J^{\prime}K^{\prime}L^{\prime} $.

For question 5:

Initial coordinates: $ J(-5, 2), K(-1, -3), L(1, 5) $
Final coordinates: $ J^{\prime}(-3, -1), K^{\prime}(1, -6), L^{\prime}(3, 2) $

Step 2: Calculate the Component Form of the Vector for Each Vertex

To find the component form of the vector that maps each vertex from its initial position to its final position, we subtract the initial coordinates from the final coordinates.

For vertex $ J $: \[ \vec{v}_J = J^{\prime} - J = (-3, -1) - (-5, 2) = (-3 + 5, -1 - 2) = (2, -3) \]

For vertex $ K $: \[ \vec{v}_K = K^{\prime} - K = (1, -6) - (-1, -3) = (1 + 1, -6 + 3) = (2, -3) \]

For vertex $ L $: \[ \vec{v}_L = L^{\prime} - L = (3, 2) - (1, 5) = (3 - 1, 2 - 5) = (2, -3) \]

Step 3: Verify Consistency of the Vector

Since the component form of the vector is the same for all vertices, we can conclude that the vector that maps $ \Delta JKL $ to $ \Delta J^{\prime}K^{\prime}L^{\prime} $ is consistent.