complete the following for each pair of vectors.
do not round any intermediate computations. round your answers to the nearest degree.

1. let $\mathbf{w} = -7\mathbf{i} + 6\mathbf{j}$ and $\mathbf{r} = 5\mathbf{i} + 4\mathbf{j}$.
(a) find the angle between $\mathbf{w}$ and $\mathbf{r}$.
(b) determine whether $\mathbf{w}$ and $\mathbf{r}$ are parallel, orthogonal, or neither.

2. let $\mathbf{t} = 4\mathbf{i} - \mathbf{j}$ and $\mathbf{v} = -8\mathbf{i} + 2\mathbf{j}$.
(a) find the angle between $\mathbf{t}$ and $\mathbf{v}$.
(b) determine whether $\mathbf{t}$ and $\mathbf{v}$ are parallel, orthogonal, or neither.

Question

complete the following for each pair of vectors.
do not round any intermediate computations. round your answers to the nearest degree.

1. let $\mathbf{w} = -7\mathbf{i} + 6\mathbf{j}$ and $\mathbf{r} = 5\mathbf{i} + 4\mathbf{j}$.
(a) find the angle between $\mathbf{w}$ and $\mathbf{r}$.
(b) determine whether $\mathbf{w}$ and $\mathbf{r}$ are parallel, orthogonal, or neither.

2. let $\mathbf{t} = 4\mathbf{i} - \mathbf{j}$ and $\mathbf{v} = -8\mathbf{i} + 2\mathbf{j}$.
(a) find the angle between $\mathbf{t}$ and $\mathbf{v}$.
(b) determine whether $\mathbf{t}$ and $\mathbf{v}$ are parallel, orthogonal, or neither.

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vector Magnitude", "Vector Dot Product", "Angle Between Vectors" ], "new_concepts": [], "current_concepts": [ "Vector Magnitude", "Vector Dot Product", "Angle Between Vectors", "Parallel and Orthogonal Vectors" ] } ## Calculate dot product and magnitudes for Question 1 Using the *Vector Dot Product* and *Vector Magnitude* knowledge points $$ \begin{aligned} &\mathbf{w} \cdot \mathbf{r} = (-7)(5) + (6)(4) = -35 + 24 = -11\ &\|\mathbf{w}\| = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}\ &\|\mathbf{r}\| = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \end{aligned} $$ ## Find the angle and relationship for Question 1 Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} &\cos\theta = \frac{\mathbf{w} \cdot \mathbf{r}}{\|\mathbf{w}\| \|\mathbf{r}\|} = \frac{-11}{\sqrt{85}\sqrt{41}} = \frac{-11}{\sqrt{3485}} \approx -0.1863\ &\theta = \arccos(-0.1863) \approx 101^{\circ} \end{aligned} $$ Since $\theta eq 0^{\circ}, 180^{\circ}$ (not parallel) and $\mathbf{w} \cdot \mathbf{r} eq 0$ (not orthogonal), the relationship is neither. ## Calculate dot product and magnitudes for Question 2 Using the *Vector Dot Product* and *Vector Magnitude* knowledge points $$ \begin{aligned} &\mathbf{t} \cdot \mathbf{v} = (4)(-8) + (-1)(2) = -32 - 2 = -34\ &\|\mathbf{t}\| = \sqrt{4^2 + (-1)^2} = \sqrt{17}\ &\|\mathbf{v}\| = \sqrt{(-8)^2 + 2^2} = \sqrt{68} = 2\sqrt{17} \end{aligned} $$ ## Find the angle and relationship for Question 2 Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} &\cos\phi = \frac{\mathbf{t} \cdot \mathbf{v}}{\|\mathbf{t}\| \|\mathbf{v}\|} = \frac{-34}{\sqrt{17} \cdot 2\sqrt{17}} = \frac{-34}{34} = -1\ &\phi = \arccos(-1) = 180^{\circ} \end{aligned} $$ Since $\phi = 180^{\circ}$, the vectors point in opposite directions and are parallel. ## Question 1 (a) Angle between $\mathbf{w}$ and $\mathbf{r}$: $101$$^{\circ}$ (b) Relationship: parallel

orthogonal

neither ## Question 2 (a) Angle between $\mathbf{t}$ and $\mathbf{v}$: $180$$^{\circ}$ (b) Relationship: parallel

orthogonal

neither

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Angle Between Vectors" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate dot product and magnitudes for Question 1

Find the angle and relationship for Question 1

Calculate dot product and magnitudes for Question 2

Find the angle and relationship for Question 2

Answer:

Question 1

Question 2