Questions: Match each vector with the accompanying description. Vectors Description A. vec(v) = <3,-4> I. Magnitude is 13 B. vec(u) = <-1, sqrt(2)> II. Magnitude is 5 C. vec(w) = <-5,-12> II. Direction is 225 degrees D. vec(x) = <-4,-4> IV. Direction is 125.64 degrees

Match each vector with the accompanying description.
Vectors Description
A. vec(v) = <3,-4> I. Magnitude is 13
B. vec(u) = <-1, sqrt(2)> II. Magnitude is 5
C. vec(w) = <-5,-12> II. Direction is 225 degrees
D. vec(x) = <-4,-4> IV. Direction is 125.64 degrees

Transcript text: Match each vector with the accompanying description. \begin{tabular}{|l|l|} \hline Vectors & Description \\ \hline A. $\vec{v}=\langle 3,-4\rangle$ & I. Magnitude is 13 \\ \hline B. $\vec{u}=\langle-1, \sqrt{2}\rangle$ & II. Magnitude is 5 \\ \hline C. $\vec{w}=\langle-5,-12\rangle$ & II. Direction is $225^{\circ}$ \\ \hline D. $\vec{x}=\langle-4,-4\rangle$ & IV. Direction is $125.64^{\circ}$ \\ \hline \end{tabular}

Solution

Solution Steps

To match each vector with its description, we need to calculate the magnitude and direction of each vector. The magnitude of a vector $\vec{v} = \langle a, b \rangle$ is given by $\sqrt{a^2 + b^2}$. The direction (angle) of a vector can be found using the arctangent function: $\theta = \tan^{-1}(b/a)$. We will calculate these for each vector and match them with the given descriptions.

Step 1: Calculate Magnitudes

For each vector, we calculate the magnitude using the formula $ \|\vec{v}\| = \sqrt{a^2 + b^2} $.

For $ \vec{v} = \langle 3, -4 \rangle $: \[ \|\vec{v}\| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
For $ \vec{u} = \langle -1, \sqrt{2} \rangle $: \[ \|\vec{u}\| = \sqrt{(-1)^2 + (\sqrt{2})^2} = \sqrt{1 + 2} = \sqrt{3} \]
For $ \vec{w} = \langle -5, -12 \rangle $: \[ \|\vec{w}\| = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
For $ \vec{x} = \langle -4, -4 \rangle $: \[ \|\vec{x}\| = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

Step 2: Calculate Directions

Next, we calculate the direction of each vector using the formula $ \theta = \tan^{-1}\left(\frac{b}{a}\right) $.

For $ \vec{v} = \langle 3, -4 \rangle $: \[ \theta = \tan^{-1}\left(\frac{-4}{3}\right) \approx -53.13^\circ \quad (\text{in the fourth quadrant, } 360^\circ - 53.13^\circ = 306.87^\circ) \]
For $ \vec{u} = \langle -1, \sqrt{2} \rangle $: \[ \theta = \tan^{-1}\left(\frac{\sqrt{2}}{-1}\right) \approx 135^\circ \]
For $ \vec{w} = \langle -5, -12 \rangle $: \[ \theta = \tan^{-1}\left(\frac{-12}{-5}\right) \approx 225^\circ \]
For $ \vec{x} = \langle -4, -4 \rangle $: \[ \theta = \tan^{-1}\left(\frac{-4}{-4}\right) = 225^\circ \]

Step 3: Match Vectors with Descriptions

Now we match the calculated magnitudes and directions with the given descriptions: