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Practice: Cross Product

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Wize University Linear Algebra Textbook > Vectors

Unit Vectors

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product

3 Activities

Practice: Cross Product

Consider uβƒ—=[βˆ’2, 3, βˆ’1]\vec{u}=\left[-2,\ 3,\ -1\right] and vβƒ—=[1, 0, βˆ’1]\vec{v}=\left[1,\ 0,\ -1\right].
How many unit vectors are perpendicular to u⃗\vec{u} and v⃗\vec{v}?

[Hint: If w⃗\vec{w} is a vector that is perpendicular to u⃗\vec{u} and v⃗\vec{v}, then any scalar multiple of w⃗\vec{w} will be a vector perpendicular to u⃗\vec{u} and v⃗\vec{v}]
More Unit Vectors Questions:
Find the unit vector in the opposite direction as 𝑣⃗ =(1,βˆ’1,0,βˆ’1,0)𝑣⃗\ =(1,βˆ’1,0,βˆ’1,0)
Find the unit vector in the same direction as vβƒ—=(1, 0, βˆ’3, 4, βˆ’2)\vec{v}=\left(1,\ 0,\ -3,\ 4,\ -2\right).
Vectors in 2D and 3D Systems (2)
Practice
Find the unit vector that points in the same direction as aβƒ— =<βˆ’10, 4, βˆ’3>\vec{a}~=<-10,~4,~-3>.
Unit Vectors
Practice: Unit Vectors
Find the value of kk such that uβƒ—=⟨k,12,βˆ’k⟩\vec u = \left\lang k, \dfrac{1}{\sqrt2}, -k \right\rang is a unit vector.
133 - FML 3 - 18.1W e.g. 26
Find on expression for a vector of unit length in the direction of v⃗=<v1,v2,v3>\bcb{\vec{v} = \left< v_1, v_2, v_3 \right>}.
Practice: Unit Vector
Practice: Unit Vectors
Find the unit vector in the same direction as vβƒ—=(1,βˆ’2,βˆ’1)\vec v = (1,-2,-1).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as iβƒ—βˆ’5kβƒ—\vec i-5\vec k.
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as i^βˆ’5k^\hat{i}-5\hat{k}.
Concept Clarifier
Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)
133 - FML 3 - 18.1W e.g. 24
If a vector v⃗\bcb{\vec{v}} has length L\bcb{\mathcal{L}}, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}.
133 - FML 3 - 18.1W e.g. 26
Find an expression for a vector of unit length in the direction of v⃗=<v1,v2,v3>\bcb{\vec{v} = \left< v_1, v_2, v_3 \right>}.
133 - FML 3 - 18.1W e.g. 48
Find a unit vector in the direction of the projection of uβƒ—=<102,112,βˆ’56>\bcb{\vec{u} = \left< 102, 112, -56 \right>} onto vβƒ—=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
133 - FML 3 - 18.1W e.g. 24
If a vector v⃗\bcb{\vec{v}} has length L\bcb{\mathcal{L}}, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 48
Find a unit vector in the direction of the projection of uβƒ—=<102,112,βˆ’56>\bcb{\vec{u} = \left< 102, 112, -56 \right>} onto vβƒ—=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
Concept Clarifier
Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)
Unit Vector Practice Question
Find the unit vector in the same direction as v⃗=(3,4,5)\vec{v}=(3,4,5)
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, βˆ’2, βˆ’1).
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, βˆ’2, βˆ’1).
Practice Question: Unit Vector
Practice Question: Unit Vector
Which one of the following is not a unit vector?
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the opposite direction as 𝑣⃗ =(1,βˆ’1,0,βˆ’1,0)𝑣⃗\ =(1,βˆ’1,0,βˆ’1,0).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, βˆ’2, βˆ’1).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as iβƒ—βˆ’5kβƒ—\vec i-5\vec k.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
A unit vector parallel to the vector vβƒ—=<4,β€‰βˆ’3>\vv = \rowvec{4}{-3} is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
A unit vector perpendicular to the vector wβƒ—=<2, 5>\vw = \rowvec{2}{5} is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
Find a vector parallel to the vector uβƒ—=<1, 4, 8>\vu = \rowvecth{1}{4}{8} whose magnitude is 1.
DIY Unit Vector
Find a vector whose length is one, that points in the same direction as the vector uβƒ—=<2, 1,β€‰βˆ’2>\vu = \rowvecth{2}{1}{-2}
Unit Vectors: Dot Product
Consider the vectors u=iβˆ’4k\bm{u}=\bm{i}-4\bm{k} and v=3iβˆ’2j+k\bm{v}=3\bm{i}-2\bm{j}+\bm{k} in R3\reals^3. Find (if possible) 3vβ‹…(kβˆ’2u)3\bm{v}\cdot(\bm{k}-2\bm{u}).
More Cross Product Questions:
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both uβƒ—=[2βˆ’13]\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix} and vβƒ—=[0βˆ’1βˆ’1]\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}.
For extra practice, check your answer!
Find vec form line thru pt ortho to 2 vec form lines
Find the equation of the line that passes through the point (1,2,βˆ’3)\left(1,2,-3\right) and is orthogonal to the lines with equations:
(x,y,z)=(βˆ’5,0,1)+t(1,0,βˆ’1)(x,y,z)= (-5,0,1)+t(1,0,-1) and (x,y,z)=(3,βˆ’1,6)+t(0,2,1)(x,y,z)=(3,-1,6)+t(0,2,1)
Which of the following vectors is orthogonal to both [1,2,3][1,2,3] and [βˆ’3,1,0][-3,1,0].
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both 𝑒⃗=(2,βˆ’1,3)𝑒⃗=(2,βˆ’1,3) and 𝑣⃗=(0,βˆ’1,βˆ’1).\vec{𝑣} = (0, βˆ’1, βˆ’1).
Consider three vectors Aβƒ—=2i^+j^,  Bβƒ—=i^βˆ’2j^\vec{A}=2\hat{i}+\hat{j},\,\,\vec{B}=\hat{i}-2\hat{j}, and Cβƒ—=βˆ’2i^+j^\vec{C}=-2\hat{i}+\hat{j}. The angle between Cβƒ—\vec{C} and Aβƒ—Γ—Bβƒ—\vec{A}\times\vec{B} is?
Evaluate 2k^Γ—(3i^Γ—2j^)2\hat{k}\times(3\hat{i}\times2\hat{j})
Practice: Cross Product
Practice: Cross Product
Recall that iβƒ—=[1, 0, 0]\vec{i}=\left[1,\ 0,\ 0\right], jβƒ—=[0, 1, 0]\vec{j}=\left[0,\ 1,\ 0\right], and kβƒ—=[0, 0, 1]\vec{k}=\left[0,\ 0,\ 1\right].
Match the following cross products with the correct result.
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both uβƒ—=[2βˆ’13]\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix} and vβƒ—=[0βˆ’1βˆ’1]\vec{v} = \begin{bmatrix} 0\\ -1\\ -1\\ \end{bmatrix}.
For extra practice, check your answer!
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find a vector orthogonal to both 𝑒⃗=(2,βˆ’1,3)𝑒⃗=(2,βˆ’1,3) and 𝑣⃗=(0,βˆ’1,βˆ’1).\vec{𝑣} = (0, βˆ’1, βˆ’1).
Practice: Orthogonal Vectors
Practice: Orthogonal Vectors
Find a vector orthogonal to both 𝑒⃗=(2,βˆ’1,3)𝑒⃗=(2,βˆ’1,3) and 𝑣⃗=(0,βˆ’1,βˆ’1).\vec{𝑣} = (0, βˆ’1, βˆ’1).
If uβƒ—=(2,0,βˆ’3), vβƒ—=(βˆ’2,2,βˆ’4)\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right) and wβƒ—=(1,2,1)\vec{w}=\left(1,2,1\right), find a vector that is orthogonal to uvβƒ—\vec{uv} and uwβƒ—\vec{uw}.
If 𝑒⃗=(2,0,βˆ’3),𝑣⃗=(βˆ’2,2,βˆ’4),𝑒⃗ = (2, 0, βˆ’3), \vec𝑣 = (βˆ’2, 2, βˆ’4), and 𝑀⃗=(1,2,1)𝑀⃗= (1,2,1), compute uβƒ— Γ— wβƒ—\vec{u}\ \times\ \vec{w}.
Give that 𝑒⃗𝑒⃗ and 𝑣⃗\vec𝑣 are vectors in R3\mathbb{R}^3, which of the following statements is/are always true?
i. 2𝑒⃗+3𝑣⃗=2𝑣⃗+3𝑒⃗2𝑒⃗ + 3\vec𝑣 = 2\vec𝑣 + 3𝑒⃗
ii. 2𝑒⃗.(βˆ’3𝑣⃗)=βˆ’2(3𝑒⃗.𝑣⃗)2𝑒⃗ . (βˆ’3\vec𝑣 ) = βˆ’2(3𝑒⃗ . \vec𝑣 )
If 𝑒⃗=(2,0,βˆ’3),𝑣⃗=(βˆ’2,2,βˆ’4),𝑒⃗ = (2, 0, βˆ’3), \vec𝑣 = (βˆ’2, 2, βˆ’4), and 𝑀⃗=(1,2,1)𝑀⃗= (1,2,1), compute uβƒ— Γ— wβƒ—\vec{u}\ \times\ \vec{w}.
If u=(2,0,βˆ’3), v=(βˆ’2,2,βˆ’4)u=\left(2,0,-3\right),\ v=\left(-2,2,-4\right) and w=(1,2,1)w=\left(1,2,1\right), find a vector that is orthogonal to uvβƒ—\vec{uv} and uwβƒ—\vec{uw}.
Practice Question: Vector Product
Practice Question: Vector Product
Given that 𝑒⃗=(1,1,βˆ’1)𝑒⃗=(1,1,βˆ’1) and 𝑣⃗=(0,βˆ’2,2),𝑣⃗=(0,βˆ’2,2), compute (𝑒⃗⋅𝑣⃗)(𝑣⃗×𝑒⃗).(𝑒⃗\cdot𝑣⃗)(𝑣⃗\times𝑒⃗).
Dot and Cross Products: Vector Norm
If uβƒ—, vβƒ—, wβƒ—  βˆˆ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Select all that are correct.