Question

in exercises 5–12, let p = (-2, 2), q = (3, 4), r = (-2, 5), and s = (2, -8). find the component form and magnitude of the vector.

Explanation:

To solve the problem, we need to know which vector we are dealing with (e.g., \(\overrightarrow{PQ}\), \(\overrightarrow{PR}\), \(\overrightarrow{PS}\), etc.). Since the problem statement in the image is cut off, I'll assume we are finding the component form and magnitude of a common vector, say \(\overrightarrow{PQ}\) (but the process is similar for any vector between two of these points).

Step 1: Recall the formula for the component form of a vector

If we have two points \( A = (x_1, y_1) \) and \( B = (x_2, y_2) \), the component form of the vector \(\overrightarrow{AB}\) is given by:

$$ \overrightarrow{AB} = \langle x_2 - x_1, y_2 - y_1 angle $$

Step 2: Recall the formula for the magnitude of a vector

For a vector \(\mathbf{v} = \langle a, b
angle\), the magnitude (or length) is given by:

$$ \|\mathbf{v}\| = \sqrt{a^2 + b^2} $$

Example: Finding \(\overrightarrow{PQ}\) (where \( P = (-2, 2) \) and \( Q = (3, 4) \))

Step 1: Component form of \(\overrightarrow{PQ}\)

Using the formula for the component form:

$$ \overrightarrow{PQ} = \langle 3 - (-2), 4 - 2 angle = \langle 5, 2 angle $$

Step 2: Magnitude of \(\overrightarrow{PQ}\)

Using the magnitude formula:

$$ \|\overrightarrow{PQ}\| = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} $$

If we instead consider \(\overrightarrow{PS}\) (where \( P = (-2, 2) \) and \( S = (2, -8) \))

Step 1: Component form of \(\overrightarrow{PS}\)

$$ \overrightarrow{PS} = \langle 2 - (-2), -8 - 2 angle = \langle 4, -10 angle $$

Step 2: Magnitude of \(\overrightarrow{PS}\)

$$ \|\overrightarrow{PS}\| = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116} = 2\sqrt{29} $$

Since the problem statement in the image is incomplete (it does not specify which vector to find), please clarify which vector (e.g., \(\overrightarrow{PQ}\), \(\overrightarrow{PR}\), \(\overrightarrow{PS}\), etc.) you need to compute, and I will provide the exact solution.

Answer:

To solve the problem, we need to know which vector we are dealing with (e.g., \(\overrightarrow{PQ}\), \(\overrightarrow{PR}\), \(\overrightarrow{PS}\), etc.). Since the problem statement in the image is cut off, I'll assume we are finding the component form and magnitude of a common vector, say \(\overrightarrow{PQ}\) (but the process is similar for any vector between two of these points).

Step 1: Recall the formula for the component form of a vector

If we have two points \( A = (x_1, y_1) \) and \( B = (x_2, y_2) \), the component form of the vector \(\overrightarrow{AB}\) is given by:

$$ \overrightarrow{AB} = \langle x_2 - x_1, y_2 - y_1 angle $$

Step 2: Recall the formula for the magnitude of a vector

For a vector \(\mathbf{v} = \langle a, b
angle\), the magnitude (or length) is given by:

$$ \|\mathbf{v}\| = \sqrt{a^2 + b^2} $$

Example: Finding \(\overrightarrow{PQ}\) (where \( P = (-2, 2) \) and \( Q = (3, 4) \))

Step 1: Component form of \(\overrightarrow{PQ}\)

Using the formula for the component form:

$$ \overrightarrow{PQ} = \langle 3 - (-2), 4 - 2 angle = \langle 5, 2 angle $$

Step 2: Magnitude of \(\overrightarrow{PQ}\)

Using the magnitude formula:

$$ \|\overrightarrow{PQ}\| = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} $$

If we instead consider \(\overrightarrow{PS}\) (where \( P = (-2, 2) \) and \( S = (2, -8) \))

Step 1: Component form of \(\overrightarrow{PS}\)

$$ \overrightarrow{PS} = \langle 2 - (-2), -8 - 2 angle = \langle 4, -10 angle $$

Step 2: Magnitude of \(\overrightarrow{PS}\)

$$ \|\overrightarrow{PS}\| = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116} = 2\sqrt{29} $$

Since the problem statement in the image is incomplete (it does not specify which vector to find), please clarify which vector (e.g., \(\overrightarrow{PQ}\), \(\overrightarrow{PR}\), \(\overrightarrow{PS}\), etc.) you need to compute, and I will provide the exact solution.