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: (\#,\#)
\[
\begin{array}{l}
\mathbf{u}=(16,-8) \\
\mathbf{v}=(-5,3) \\
\mathbf{u}+\mathbf{v}=?
\end{array}
\]
Enter answer:
Vector Subtraction
Given vectors \(\vec{a} = [3, -1, 2]\) and \(\vec{b} = [2, 4, -1]\), find the vector \(\vec{a} - \vec{b}\).
Vector Multiplication by a Scalar
For $\mathbf{u}=\langle 3,-1\rangle, \mathbf{v}=\langle 3,1\rangle$, and $\mathbf{w}=\langle 1,3\rangle$, evaluate the expression
$(4 u) \cdot v$
$(4 u) \cdot v=\square($ Simplify your answer.)
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 $\vec{v}-(1,2,1)$
\[
\left\{\begin{array}{l}
z=3+t \\
y=4+2 t, t \in \mathbb{R}^{*} \\
z=1+t
\end{array}\right.
\]
\[
\left\{\begin{array}{l}
z=3+t \\
y=4+2 t, 0 \leq t \leq 1 \\
z=1+t
\end{array}\right.
\]
\[
\left\{\begin{array}{l}
z=3+t \\
y=4+2 t, t \geq 0 \\
z=1+t
\end{array}\right.
\]
Finding the Norm in Real Vector Space
Find the norm of the vector \( v = [3, -4, 12] \) in real vector space.
Finding the Length
Find the length of the vector \( \vec{v} = 3\hat{i} - 4\hat{j} + 2\hat{k} \)
Finding the Direction Angle of the Vector
Find the direction angle of the vector \(\vec{v} = 2\hat{i} + 3\hat{j}\)
Finding the Dot Product of Vectors
Find $\mathbf{u} \cdot \mathbf{v}$, where $\theta$ is the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$.
\[
\|\mathbf{u}\|=7,\|\mathbf{v}\|=8, \theta=\frac{\pi}{3}
\]
Determining if Vectors are Orthogonal
Determine whether the vectors \( \mathbf{A} = [1, 2, 3] \) and \( \mathbf{B} = [4, -2, 0] \) are orthogonal.
Finding the Distance Between the Vectors
Calcule a distância entre as retas:
\[
\begin{array}{l}
r:(x, y, z)=(0,7,-4)+t_{r}(4,0,7) \\
\text { e } \\
s:(x, y, z)=(4,0,4)+t_{s}(-7,0,0)
\end{array}
\]
Finding a Unit Vector in the Same Direction as the Given Vector
Find a unit vector in the same direction as the given vector \( \vec{v} = 3\hat{i} - 4\hat{j} + 2\hat{k} \).
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.
Finding the Angle Between Two Vectors Using the Dot Product
Given $\mathbf{v}=-7 \mathbf{i}-6 \mathbf{j}$ and $\mathbf{w}=5 \mathbf{i}-\mathbf{j}$, find the angle between $\mathbf{v}$ and $\mathbf{w}$.
Finding the Projection of One Vector Onto another Vector
Find the scalar magnitude of the projection of $2 i-3 j+4 k$ on $i-j$ Express as a Cartesian vector: $\vec{a}=(2,-3,4) \quad$ and $\quad \vec{b}=(1,-1,0)$
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.