Vectors

Vector Addition

Give the component form of the resultant vector in the following. NOTE: Answer must be typed in using the following format -- including the parentheses: (\#,\#)

\begin{array}{l} u = (16, - 8) \\ v = (- 5, 3) \\ u + v = ? \end{array}

Enter answer:

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Vector Subtraction

Given vectors

\vec{a} = [3, - 1, 2]

and

\vec{b} = [2, 4, - 1]

, find the vector

\vec{a} - \vec{b}

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Vector Multiplication by a Scalar

For

u = ⟨ 3, - 1 ⟩, v = ⟨ 3, 1 ⟩

, and

w = ⟨ 1, 3 ⟩

, evaluate the expression

(4 u) \cdot v

(4 u) \cdot v = ◻ (

Simplify your answer.)

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Finding the Variables

QUESIONS Identifying the Equation of a Line Segment in 3D Choose one

\cdot 5

points Identify the equation of aline-segment with origin

{\vec{S}}_{0} - (3, 4, 1)

and direction

\vec{v} - (1, 2, 1)

{\begin{cases} z = 3 + t \\ y = 4 + 2 t, t \in R^{*} \\ z = 1 + t \end{cases}

{\begin{cases} z = 3 + t \\ y = 4 + 2 t, 0 \leq t \leq 1 \\ z = 1 + t \end{cases}

{\begin{cases} z = 3 + t \\ y = 4 + 2 t, t \geq 0 \\ z = 1 + t \end{cases}

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Finding the Norm in Real Vector Space

Find the norm of the vector

v = [3, - 4, 12]

in real vector space.

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Finding the Length

Find the length of the vector

\vec{v} = 3 \hat{i} - 4 \hat{j} + 2 \hat{k}

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Finding the Direction Angle of the Vector

Find the direction angle of the vector

\vec{v} = 2 \hat{i} + 3 \hat{j}

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Finding the Dot Product of Vectors

Find

u \cdot v

, where

θ

is the angle between the vectors

u

and

v

‖ u ‖ = 7, ‖ v ‖ = 8, θ = \frac{π}{3}

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Determining if Vectors are Orthogonal

Determine whether the vectors

A = [1, 2, 3]

and

B = [4, - 2, 0]

are orthogonal.

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Finding the Distance Between the Vectors

Calcule a distância entre as retas:

\begin{array}{l} r : (x, y, z) = (0, 7, - 4) + t_{r} (4, 0, 7) \\ e \\ s : (x, y, z) = (4, 0, 4) + t_{s} (- 7, 0, 0) \end{array}

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Finding a Unit Vector in the Same Direction as the Given Vector

Find a unit vector in the same direction as the given vector

\vec{v} = 3 \hat{i} - 4 \hat{j} + 2 \hat{k}

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Finding the Angle Between Two Vectors Using the Cross Product

Find the angle between vectors

\vec{u} = (3, 4, 0)

and

\vec{v} = (2, 1, - 2)

using the cross product.

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Finding the Angle Between Two Vectors Using the Dot Product

Given

v = - 7 i - 6 j

and

w = 5 i - j

, find the angle between

v

and

w

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Finding the Projection of One Vector Onto another Vector

Find the scalar magnitude of the projection of

2 i - 3 j + 4 k

i - j

Express as a Cartesian vector:

\vec{a} = (2, - 3, 4)

and

\vec{b} = (1, - 1, 0)

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Finding an Orthonormal Basis by Gram-Schmidt Method

Given vectors v1 = [1, 2, 3] and v2 = [4, 5, 6]. Find an orthonormal basis using the Gram-Schmidt process.

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Vector Addition

Vector Subtraction

Vector Multiplication by a Scalar

Finding the Variables

Finding the Norm in Real Vector Space

Finding the Length

Finding the Direction Angle of the Vector

Finding the Dot Product of Vectors

Determining if Vectors are Orthogonal

Finding the Distance Between the Vectors

Finding a Unit Vector in the Same Direction as the Given Vector

Finding the Angle Between Two Vectors Using the Cross Product

Finding the Angle Between Two Vectors Using the Dot Product

Finding the Projection of One Vector Onto another Vector

Finding an Orthonormal Basis by Gram-Schmidt Method

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