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

1. let $\mathbf{s} = -4\mathbf{i} - 5\mathbf{j}$ and $\mathbf{t} = -8\mathbf{i} + 5\mathbf{j}$.
(a) find the angle between $\mathbf{s}$ and $\mathbf{t}$.
$\square^{\circ}$
(b) determine whether $\mathbf{s}$ and $\mathbf{t}$ are parallel, orthogonal, or neither.
parallel
orthogonal
neither

2. let $\mathbf{u} = 2\mathbf{i} + \mathbf{j}$ and $\mathbf{w} = -\mathbf{i} + 2\mathbf{j}$.
(a) find the angle between $\mathbf{u}$ and $\mathbf{w}$.
$\square^{\circ}$
(b) determine whether $\mathbf{u}$ and $\mathbf{w}$ are parallel, orthogonal, or neither.
parallel
orthogonal
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 $\mathbf{s} = -4\mathbf{i} - 5\mathbf{j}$ and $\mathbf{t} = -8\mathbf{i} + 5\mathbf{j}$.
(a) find the angle between $\mathbf{s}$ and $\mathbf{t}$.
$\square^{\circ}$
(b) determine whether $\mathbf{s}$ and $\mathbf{t}$ are parallel, orthogonal, or neither.
parallel
orthogonal
neither

2. let $\mathbf{u} = 2\mathbf{i} + \mathbf{j}$ and $\mathbf{w} = -\mathbf{i} + 2\mathbf{j}$.
(a) find the angle between $\mathbf{u}$ and $\mathbf{w}$.
$\square^{\circ}$
(b) determine whether $\mathbf{u}$ and $\mathbf{w}$ are parallel, orthogonal, or neither.
parallel
orthogonal
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 Dot Product", "Vector Magnitude", "Angle Between Vectors" ], "new_concepts": [], "current_concepts": [ "Vector Dot Product", "Vector Magnitude", "Angle Between Vectors", "Orthogonal Vectors", "Parallel Vectors" ] } ## Calculate dot product and magnitudes for Question 1 Using the *Vector Dot Product* and *Vector Magnitude* knowledge points $$ \begin{aligned} \mathbf{s} \cdot \mathbf{t} &= (-4)(-8) + (-5)(5) = 32 - 25 = 7 \ \|\mathbf{s}\| &= \sqrt{(-4)^2 + (-5)^2} = \sqrt{16 + 25} = \sqrt{41} \ \|\mathbf{t}\| &= \sqrt{(-8)^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89} \end{aligned} $$ ## Find angle and relationship for Question 1 Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} \cos\theta &= \frac{\mathbf{s} \cdot \mathbf{t}}{\|\mathbf{s}\| \|\mathbf{t}\|} = \frac{7}{\sqrt{41}\sqrt{89}} = \frac{7}{\sqrt{3649}} \approx 0.11588 \ \theta &= \arccos(0.11588) \approx 83.34^\circ \approx 83^\circ \end{aligned} $$ Since the angle is not $0^\circ$, $180^\circ$, or $90^\circ$, the vectors are neither parallel nor orthogonal. ## Calculate dot product and magnitudes for Question 2 Using the *Vector Dot Product* and *Vector Magnitude* knowledge points $$ \begin{aligned} \mathbf{u} \cdot \mathbf{w} &= (2)(-1) + (1)(2) = -2 + 2 = 0 \ \|\mathbf{u}\| &= \sqrt{2^2 + 1^2} = \sqrt{5} \ \|\mathbf{w}\| &= \sqrt{(-1)^2 + 2^2} = \sqrt{5} \end{aligned} $$ ## Find angle and relationship for Question 2 Using the *Angle Between Vectors* knowledge point $$ \begin{aligned} \cos\theta &= \frac{\mathbf{u} \cdot \mathbf{w}}{\|\mathbf{u}\| \|\mathbf{w}\|} = \frac{0}{\sqrt{5}\sqrt{5}} = 0 \ \theta &= \arccos(0) = 90^\circ \end{aligned} $$ Since the dot product is $0$, the angle is exactly $90^\circ$, which means the vectors are orthogonal. ## Question 1 (a) The angle between $\mathbf{s}$ and $\mathbf{t}$ is $83$$^\circ$. (b) Determine whether $\mathbf{s}$ and $\mathbf{t}$ are parallel, orthogonal, or neither. parallel

orthogonal

neither ## Question 2 (a) Find the angle between $\mathbf{u}$ and $\mathbf{w}$. The angle between $\mathbf{u}$ and $\mathbf{w}$ is $90$$^\circ$. (b) Determine whether $\mathbf{u}$ and $\mathbf{w}$ are parallel, orthogonal, or neither. 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 angle and relationship for Question 1

Calculate dot product and magnitudes for Question 2

Find angle and relationship for Question 2

Answer:

Question 1

Question 2