Answer :
To find the magnitude and direction of the vector [tex]\( \mathbf{w} = (-9, -19) \)[/tex], let's go through the calculations step by step.
### Step 1: Calculate the Magnitude
The magnitude [tex]\( \| \mathbf{w} \| \)[/tex] of a vector [tex]\( \mathbf{w} = (x, y) \)[/tex] is given by the formula:
[tex]\[ \| \mathbf{w} \| = \sqrt{x^2 + y^2} \][/tex]
For the vector [tex]\( \mathbf{w} = (-9, -19) \)[/tex]:
[tex]\[ x = -9 \][/tex]
[tex]\[ y = -19 \][/tex]
Now, substitute the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the magnitude formula:
[tex]\[ \| \mathbf{w} \| = \sqrt{(-9)^2 + (-19)^2} \][/tex]
[tex]\[ \| \mathbf{w} \| = \sqrt{81 + 361} \][/tex]
[tex]\[ \| \mathbf{w} \| = \sqrt{442} \][/tex]
[tex]\[ \| \mathbf{w} \| \approx 21.024 \][/tex]
### Step 2: Calculate the Direction (Angle)
The direction [tex]\( \theta \)[/tex] of vector [tex]\( \mathbf{w} = (x, y) \)[/tex] is given by the formula:
[tex]\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \][/tex]
For the vector [tex]\( \mathbf{w} = (-9, -19) \)[/tex]:
[tex]\[ x = -9 \][/tex]
[tex]\[ y = -19 \][/tex]
Now, substitute the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the direction formula (and we need to ensure the angle is in the correct quadrant). We'll use the arctangent function that takes into account the signs of both arguments to determine the correct quadrant, commonly known as atan2:
[tex]\[ \theta = \tan^{-1} \left( \frac{-19}{-9} \right) \][/tex]
In radians:
[tex]\[ \theta = \tan^{-1} \left( \frac{19}{9} \right) \][/tex]
After converting to degrees (and accounting for the signs which place it in the correct quadrant):
[tex]\[ \theta \approx -115.346^{\circ} \][/tex]
### Step 3: Match the Calculated Values with Given Choices
Given choices are:
1. [tex]\( \| \mathbf{w} \| = 5.099 \)[/tex] ; [tex]\( \theta = 205.346^{\circ} \)[/tex]
2. [tex]\( \| \mathbf{w} \| = 21.024 \)[/tex] ; [tex]\( \theta = 244.654^{\circ} \)[/tex]
3. [tex]\( \| \mathbf{w} \| = 18.047 \)[/tex] ; [tex]\( \theta = 25.346^{\circ} \)[/tex]
4. [tex]\( \| \mathbf{w} \| = 25.298 \)[/tex] ; [tex]\( \theta = 64.654^{\circ} \)[/tex]
From our calculations:
[tex]\[ \| \mathbf{w} \| \approx 21.024 \][/tex]
[tex]\[ \theta \approx -115.346^{\circ} \][/tex]
None of the provided choices exactly match the calculated direction. Therefore, we conclude that none of the given options are correct based on our step-by-step solution.
### Step 1: Calculate the Magnitude
The magnitude [tex]\( \| \mathbf{w} \| \)[/tex] of a vector [tex]\( \mathbf{w} = (x, y) \)[/tex] is given by the formula:
[tex]\[ \| \mathbf{w} \| = \sqrt{x^2 + y^2} \][/tex]
For the vector [tex]\( \mathbf{w} = (-9, -19) \)[/tex]:
[tex]\[ x = -9 \][/tex]
[tex]\[ y = -19 \][/tex]
Now, substitute the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the magnitude formula:
[tex]\[ \| \mathbf{w} \| = \sqrt{(-9)^2 + (-19)^2} \][/tex]
[tex]\[ \| \mathbf{w} \| = \sqrt{81 + 361} \][/tex]
[tex]\[ \| \mathbf{w} \| = \sqrt{442} \][/tex]
[tex]\[ \| \mathbf{w} \| \approx 21.024 \][/tex]
### Step 2: Calculate the Direction (Angle)
The direction [tex]\( \theta \)[/tex] of vector [tex]\( \mathbf{w} = (x, y) \)[/tex] is given by the formula:
[tex]\[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \][/tex]
For the vector [tex]\( \mathbf{w} = (-9, -19) \)[/tex]:
[tex]\[ x = -9 \][/tex]
[tex]\[ y = -19 \][/tex]
Now, substitute the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the direction formula (and we need to ensure the angle is in the correct quadrant). We'll use the arctangent function that takes into account the signs of both arguments to determine the correct quadrant, commonly known as atan2:
[tex]\[ \theta = \tan^{-1} \left( \frac{-19}{-9} \right) \][/tex]
In radians:
[tex]\[ \theta = \tan^{-1} \left( \frac{19}{9} \right) \][/tex]
After converting to degrees (and accounting for the signs which place it in the correct quadrant):
[tex]\[ \theta \approx -115.346^{\circ} \][/tex]
### Step 3: Match the Calculated Values with Given Choices
Given choices are:
1. [tex]\( \| \mathbf{w} \| = 5.099 \)[/tex] ; [tex]\( \theta = 205.346^{\circ} \)[/tex]
2. [tex]\( \| \mathbf{w} \| = 21.024 \)[/tex] ; [tex]\( \theta = 244.654^{\circ} \)[/tex]
3. [tex]\( \| \mathbf{w} \| = 18.047 \)[/tex] ; [tex]\( \theta = 25.346^{\circ} \)[/tex]
4. [tex]\( \| \mathbf{w} \| = 25.298 \)[/tex] ; [tex]\( \theta = 64.654^{\circ} \)[/tex]
From our calculations:
[tex]\[ \| \mathbf{w} \| \approx 21.024 \][/tex]
[tex]\[ \theta \approx -115.346^{\circ} \][/tex]
None of the provided choices exactly match the calculated direction. Therefore, we conclude that none of the given options are correct based on our step-by-step solution.
