Define the points $$ Q(3,0) $$ and $$ R(-5,6) $$. Carry out the following calculation. Find the unit vector with the same direction as $$ \overrightarrow{Q R} $$.

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UpStudy AI · Accepted Answer

To find the unit vector in the direction of the vector $$\overrightarrow{QR}$$, follow these steps:

1. **Determine the Vector $$\overrightarrow{QR}$$:**
   
   Given points:
   $$
   Q(3, 0) \quad \text{and} \quad R(-5, 6)
   $$
   
   The vector $$\overrightarrow{QR}$$ is calculated by subtracting the coordinates of $$Q$$ from $$R$$:
   $$
   \overrightarrow{QR} = R - Q = (-5 - 3, 6 - 0) = (-8, 6)
   $$

2. **Calculate the Magnitude of $$\overrightarrow{QR}$$:**
   
   The magnitude $$|\overrightarrow{QR}|$$ is given by:
   $$
   |\overrightarrow{QR}| = \sqrt{(-8)^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   $$

3. **Find the Unit Vector:**
   
   A unit vector in the direction of $$\overrightarrow{QR}$$ is obtained by dividing each component of $$\overrightarrow{QR}$$ by its magnitude:
   $$
   \mathbf{u} = \left( \frac{-8}{10}, \frac{6}{10} \right) = \left( -\frac{4}{5}, \frac{3}{5} \right)
   $$

So, the unit vector in the direction of $$\overrightarrow{QR}$$ is:
   $$
   \mathbf{u} = \left( -\frac{4}{5}, \frac{3}{5} \right)
   $$

**Final Answer:**

After simplifying, a unit vector in the direction of QR is ( –4⁄5,  3⁄5 ).
A unit vector in the direction of QR is $$\left( -\frac{4}{5}, \frac{3}{5} \right)$$.

Define the points and . Carry out the following calculation.
Find the unit vector with the same direction as .

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Define the points and . Carry out the following calculation. Find the unit vector with the same direction as .