Define the points $$ P(-4,-2) $$ and $$ Q(3,-4) $$. Carry out the following calculation. Find two vectors parallel to $$ \overrightarrow{Q P} $$ with length 2 . The parallel vector of length 2 with the same direction is $$ \left\langle\frac{-14 \sqrt{53}}{53}, \frac{4 \sqrt{53}}{53}\right) $$. (Type exact answers, using radicals as needed.) The parallel vector of length 2 with the opposite direction is $$ \langle\square, \square $$. (Type exact answers, using radicals as needed.)

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UpStudy AI · Accepted Answer

To find two vectors parallel to $$ \overrightarrow{QP} $$ with a length of 2, let's follow the steps below.

### Step 1: Determine the Vector $$ \overrightarrow{QP} $$

Given the points:
- $$ P(-4, -2) $$
- $$ Q(3, -4) $$

The vector $$ \overrightarrow{QP} $$ is calculated as:
$$
\overrightarrow{QP} = P - Q = (-4 - 3, -2 - (-4)) = (-7, 2)
$$

### Step 2: Calculate the Magnitude of $$ \overrightarrow{QP} $$

$$
\| \overrightarrow{QP} \| = \sqrt{(-7)^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}
$$

### Step 3: Find the Unit Vector in the Direction of $$ \overrightarrow{QP} $$

$$
\text{Unit vector} = \left\langle \frac{-7}{\sqrt{53}}, \frac{2}{\sqrt{53}} \right\rangle
$$

### Step 4: Scale the Unit Vector to Length 2

$$
\text{Vector with length 2 (same direction)} = 2 \times \left\langle \frac{-7}{\sqrt{53}}, \frac{2}{\sqrt{53}} \right\rangle = \left\langle \frac{-14}{\sqrt{53}}, \frac{4}{\sqrt{53}} \right\rangle
$$

To rationalize the denominators:
$$
\left\langle \frac{-14}{\sqrt{53}} \times \frac{\sqrt{53}}{\sqrt{53}}, \frac{4}{\sqrt{53}} \times \frac{\sqrt{53}}{\sqrt{53}} \right\rangle = \left\langle \frac{-14\sqrt{53}}{53}, \frac{4\sqrt{53}}{53} \right\rangle
$$

This matches the vector you provided:
$$
\left\langle \frac{-14\sqrt{53}}{53}, \frac{4\sqrt{53}}{53} \right\rangle
$$

### Step 5: Find the Vector with Length 2 in the Opposite Direction

To get the vector in the opposite direction, simply multiply the original vector by -1:
$$
\left\langle \frac{14\sqrt{53}}{53}, \frac{-4\sqrt{53}}{53} \right\rangle
$$

### Final Answer

The parallel vector of length 2 with the opposite direction is
$$
\boxed{\left\langle\, \dfrac{14\,\sqrt{53}}{53},\;\dfrac{-4\,\sqrt{53}}{53}\,\right\rangle}
$$
The parallel vector of length 2 in the opposite direction is $$ \left\langle \frac{14\sqrt{53}}{53}, \frac{-4\sqrt{53}}{53} \right\rangle $$.

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