Question 1
Let $\mathbf{v}=\left(\begin{array}{c}1 \ -4 \ 1\end{array} ight)$ and $\mathbf{w}=\left(\begin{array}{c}1 \ -1 \ -3\end{array} ight) \in \mathbb{R}^{3}$.
a)

Calculate v $\cdot \mathbf{w}$. $\square$
b)

What is the angle $ heta$ between $\mathbf{v}$ and $\mathbf{w}$ in radians? $\square$
Round your answer to 3 decimal places.
All answers saved
2 marks

Question

Question 1
Let $\mathbf{v}=\left(\begin{array}{c}1 \ -4 \ 1\end{array}ight)$ and $\mathbf{w}=\left(\begin{array}{c}1 \ -1 \ -3\end{array}ight) \in \mathbb{R}^{3}$.
a)

Calculate v $\cdot \mathbf{w}$. $\square$
b)

What is the angle $	heta$ between $\mathbf{v}$ and $\mathbf{w}$ in radians? $\square$
Round your answer to 3 decimal places.
All answers saved
2 marks

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To calculate the dot product of vectors $ \mathbf{v} = (1, -4, 1) $ and $ \mathbf{w} = (1, -1, -3) $, we use the formula: $ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 + v_3 w_3 $. ‖ ‖ step 2 ⋮ Substituting the values, we have: $ 1 \cdot 1 + (-4) \cdot (-1) + 1 \cdot (-3) = 1 + 4 - 3 $. ‖ ‖ step 3 ⋮ Simplifying this gives: $ 1 + 4 - 3 = 2 $. ‖ ‖ step 4 ⋮ The angle $ \theta $ between the vectors can be calculated using the formula: $ \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \cdot ||\mathbf{w}||} $. ‖ ‖ step 5 ⋮ First, we find the magnitudes: $ ||\mathbf{v}|| = \sqrt{1^2 + (-4)^2 + 1^2} = \sqrt{1 + 16 + 1} = \sqrt{18} $ and $ ||\mathbf{w}|| = \sqrt{1^2 + (-1)^2 + (-3)^2} = \sqrt{1 + 1 + 9} = \sqrt{11} $. ‖ ‖ step 6 ⋮ Now substituting into the cosine formula: $ \cos(\theta) = \frac{2}{\sqrt{18} \cdot \sqrt{11}} $. ‖ ‖ step 7 ⋮ Calculating $ \sqrt{18} \cdot \sqrt{11} = \sqrt{198} $, thus $ \cos(\theta) = \frac{2}{\sqrt{198}} $. ‖ ‖ step 8 ⋮ Finally, to find $ \theta $, we take the inverse cosine: $ \theta = \cos^{-1}\left(\frac{2}{\sqrt{198}}\right) $. Rounding to three decimal places gives the final angle. ‖ ∻Answer∻ ⚹ Dot product: 2; Angle $ \theta $: [Insert angle in radians rounded to three decimal places] ⚹ ∻Key Concept∻ ⚹ The dot product and angle between vectors are fundamental concepts in vector algebra. ⚹ ∻Explanation∻ ⚹ The dot product provides a measure of how parallel two vectors are, while the angle gives insight into their directional relationship. ⚹◊What is the formula to calculate the dot product of two vectors?⍭ Generate me a similar question◊

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