Given points P(-9, 3) and Q(-4, 8), find the magnitude and the unit vector.
Understand the Problem
The question asks us to find the magnitude and the unit vector between two points P(-9, 3) and Q(-4, 8). To find the magnitude, we calculate the distance between the two points. To find the unit vector, we normalize the vector connecting the two points by dividing it by its magnitude.
Answer
Magnitude: $5\sqrt{2}$ Unit vector: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
Answer for screen readers
Magnitude: $5\sqrt{2}$
Unit vector: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
Steps to Solve
- Find the vector between points P and Q
To find the vector $\overrightarrow{PQ}$, subtract the coordinates of point $P$ from the coordinates of point $Q$.
$\overrightarrow{PQ} = Q - P = (-4, 8) - (-9, 3) = (-4 - (-9), 8 - 3) = (5, 5)$
- Calculate the magnitude of vector $\overrightarrow{PQ}$
The magnitude (or length) of a vector $(x, y)$ is given by $\sqrt{x^2 + y^2}$. Therefore, the magnitude of $\overrightarrow{PQ}$ is:
$|\overrightarrow{PQ}| = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$
- Calculate the unit vector
A unit vector is a vector with a magnitude of 1. To find the unit vector $\hat{u}$ in the direction of $\overrightarrow{PQ}$, divide the vector $\overrightarrow{PQ}$ by its magnitude.
$\hat{u} = \frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|} = \frac{(5, 5)}{5\sqrt{2}} = (\frac{5}{5\sqrt{2}}, \frac{5}{5\sqrt{2}}) = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$
Rationalizing the denominator, we get:
$\hat{u} = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
Magnitude: $5\sqrt{2}$
Unit vector: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
More Information
The unit vector gives the direction of the vector $\overrightarrow{PQ}$, and its magnitude is always 1.
Tips
A common mistake is to subtract the coordinates in the wrong order (i.e., $P - Q$ instead of $Q - P$). This will result in a vector pointing in the opposite direction, but the magnitude will still be correct. However, the unit vector will also be pointing in the opposite direction, leading to an incorrect answer.
Another common mistake is to not rationalize the denominator when finding the unit vector. While $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ is mathematically correct, it's generally preferred to express the unit vector with a rationalized denominator.
Finally, a common mistake is to incorrectly calculate the magnitude of the vector. Ensure you square each component, add them, and then take the square root of the sum.
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