Find the vector joining the points P(2, 2, 0) and α(1, -2, -4) directed from P to α.
Understand the Problem
The question is asking for the computation of a vector joining two points P and α using their coordinates. The focus is on finding the vector and identifying its components.
Answer
The vector joining the points P(2, 2, 0) and α(1, -2, -4) is $$ \vec{Pα} = -\hat{i} - 4\hat{j} - 4\hat{k} $$
Answer for screen readers
The vector joining the points P(2, 2, 0) and α(1, -2, -4) is $$ \vec{Pα} = -\hat{i} - 4\hat{j} - 4\hat{k} $$
Steps to Solve
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Identify the points
The given points are P(2, 2, 0) and α(1, -2, -4).
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Set up the formula for the vector
The vector $\vec{Pα}$ can be found using the formula: $$ \vec{Pα} = \alpha - P $$
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Calculate the vector components
Substituting the coordinates into the formula: $$ \vec{Pα} = (1 - 2)\hat{i} + (-2 - 2)\hat{j} + (-4 - 0)\hat{k} $$
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Simplify the components
This simplifies to: $$ \vec{Pα} = -1\hat{i} - 4\hat{j} - 4\hat{k} $$
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Final vector representation
Thus, the vector joining P to α is: $$ \vec{Pα} = -\hat{i} - 4\hat{j} - 4\hat{k} $$
The vector joining the points P(2, 2, 0) and α(1, -2, -4) is $$ \vec{Pα} = -\hat{i} - 4\hat{j} - 4\hat{k} $$
More Information
This vector shows the direction and distance from point P to point α in three-dimensional space. Each component represents how much to move along the x, y, and z axes to go from P to α.
Tips
- Confusing the direction of the vector; ensure to subtract the coordinates in the correct order (from α to P).
- Forgetting to include the correct unit vectors ($\hat{i}$, $\hat{j}$, $\hat{k}$) while representing the final answer.
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