Practice Question: Vector Product

Dot Product

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Wize University Linear Algebra Textbook > Products of Vectors

Cross Product

3 Activities

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Practice Question: Vector Product
Given that 𝑢⃗=(1,1,−1)𝑢⃗=(1,1,−1)u⃗=(1,1,−1) and 𝑣⃗=(0,−2,2),𝑣⃗=(0,−2,2),v⃗=(0,−2,2), compute (𝑢⃗⋅𝑣⃗)(𝑣⃗×𝑢⃗).(𝑢⃗\cdot𝑣⃗)(𝑣⃗\times𝑢⃗).(u⃗⋅v⃗)(v⃗×u⃗).

I have answered this question

133 - FML 3 - 18.1W e.g. 20

For two vectors v⃗\bcb{\vv}v and w⃗\bcb{\vw}w ∈  I ⁣Rn\bcb{\in}\; \R{n}∈IRn, v⃗⋅w⃗:=‾\bcb{\vv \cdot \vw :=\underline{\qquad\qquad\qquad}}v⋅w:=​.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 40_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$

 If u⃗=<1,−2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>}u=⟨1,−2,3⟩ and v⃗=<2, −1, 3>\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}v=⟨2,−1,3⟩, find (2u⃗)⋅(3v⃗)\bcb{(2\vec{u})\cdot (3\vec{v})}(2u)⋅(3v). 

Example: Orthogonal Vectors

Example:
Find all values of kkk such that the vectorsv⃗=(k, 1)\vec{v}=\left(k,\ 1\right)v=(k, 1) and u⃗=(k,−1)\vec{u}=\left(k,-1\right)u=(k,−1) are orthogonal.

Practice Question: Orthogonal Vectors

Practice Question: Orthogonal Vectors
Find the value(s) of ccc such that u⃗=(−1,c,2) \vec{u}=(-1,c,2)\ u=(−1,c,2)  and v⃗=(−7, c, −4)\vec{v}=\left(-7,\ c,\ -4\right)v=(−7, c, −4) are orthogonal.

Practice Question: Orthogonal Vectors

Practice Question: Orthogonal Vectors
Find the value(s) of ccc such that u⃗=(−1,c,2) \vec{u}=(-1,c,2)\ u=(−1,c,2)  and v⃗=(−7, c, −4)\vec{v}=\left(-7,\ c,\ -4\right)v=(−7, c, −4) are orthogonal.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 52

 If ∣u⃗∣=2\bcb{|{ \vec{u}| } = 2}∣u∣=2 and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}∣v∣=3 and u⃗\bcb{\vec{u}}u has the same direction as v⃗\bcb{\vec{v}}v then (3u⃗)⋅(2v⃗)\bcb{(3\vec{u}) \cdot (2\vec{v})}(3u)⋅(2v) is:

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock ✓}$ 133 - FML 3 - 18.1W e.g. 46_$\tkcth{Mock F}\tkct{?}$_$\tkct{vid soln only}$

If ∣u⃗∣=2\bcb{|{\vec{u}|} = 2}∣u∣=2 and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}∣v∣=3, where u⃗⋅v⃗=2\bcb{\vec{u} \cdot \vec{v} = 2}u⋅v=2, find ∣u⃗−v⃗∣\bcb{|{\vec{u} - \vec{v} }|}∣u−v∣.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 54

Find the value of k\bcb{k}k such that the vector u⃗=<1,k,3>\bcb{\vec{u} = \left< 1, k, 3 \right>}u=⟨1,k,3⟩ is perpendicular to the vector v⃗=<2,−1,k>\bcb{\vec{v} = \left< 2, -1, k\right>}v=⟨2,−1,k⟩. 

If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4),u⃗=(2,0,−3),v=(−2,2,−4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1)w⃗=(1,2,1), compute u⃗ ∙ v⃗\vec{u}\ \bullet\ \vec{v}u ∙ v.

Find the value(s) of kkk(if any) such that the vectors u→=(1,k,1,0,−1)\overrightarrow{u}=\left(1,k,1,0,-1\right)u=(1,k,1,0,−1)and v→=(k,2,0,3,−4)\overrightarrow{v}=\left(k,2,0,3,-4\right)v=(k,2,0,3,−4)are orthogonal.

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ | Basic dot product.

The dot product v⃗⋅w⃗\vv \cdot \vwv⋅w, for v⃗=<−1, 1, −1>\vv = \rowvecth{-1}{1}{-1}v=⟨−1,1,−1⟩ and w⃗=<−2, −2, 3>\vw = \rowvecth{-2}{-2}{3}w=⟨−2,−2,3⟩, is equal to:

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

A unit vector perpendicular to the vector w⃗=<2, 5>\vw = \rowvec{2}{5}w=⟨2,5⟩ is:

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

A vector perpendicular to the vector connecting the points P1(2,2,2)P_1(2,2,2)P1​(2,2,2) and P2(1,−1,1)P_2(1,-1,1)P2​(1,−1,1) is: 

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

A vector perpendicular to the vector connecting the points P1(0,1,−2)P_1(0,1,-2)P1​(0,1,−2) and P2(−2,0,1)P_2(-2,0,1)P2​(−2,0,1) is: 

Dot and Cross Products: Vector Norm

If u⃗, v⃗, w⃗  ∈ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3u, v, w  ∈ R3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Select all that are correct.

Unit Vectors: Dot Product

Consider the vectors u=i−4k\bm{u}=\bm{i}-4\bm{k}u=i−4k and v=3i−2j+k\bm{v}=3\bm{i}-2\bm{j}+\bm{k}v=3i−2j+k in R3\reals^3R3. Find (if possible) 3v⋅(k−2u)3\bm{v}\cdot(\bm{k}-2\bm{u})3v⋅(k−2u).

Cross Product

Practice: Orthogonal Vectors
Find a vector orthogonal to both u⃗=[2−13]\vec u 
=
\begin{bmatrix}
  2\\ -1\\ 3
\end{bmatrix}u=​2−13​​ and v⃗=[0−1−1]\vec{v} = 
\begin{bmatrix}
  0\\ -1\\ -1\\
\end{bmatrix}v=​0−1−1​​. 
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,−3)\left(1,2,-3\right)(1,2,−3) and is orthogonal to the lines with equations:
 (x,y,z)=(−5,0,1)+t(1,0,−1)(x,y,z)= (-5,0,1)+t(1,0,-1)(x,y,z)=(−5,0,1)+t(1,0,−1) and (x,y,z)=(3,−1,6)+t(0,2,1)(x,y,z)=(3,-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][1,2,3] and [−3,1,0][-3,1,0][−3,1,0].

Practice Question: Orthogonal Vectors

Practice Question: Orthogonal Vectors
Find a vector orthogonal to both 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).v=(0,−1,−1).

Consider three vectors A⃗=2i^+j^,  B⃗=i^−2j^\vec{A}=2\hat{i}+\hat{j},\,\,\vec{B}=\hat{i}-2\hat{j}A=2i^+j^​,B=i^−2j^​, and C⃗=−2i^+j^\vec{C}=-2\hat{i}+\hat{j}C=−2i^+j^​. The angle between C⃗\vec{C}C and A⃗×B⃗\vec{A}\times\vec{B}A×B is?

Evaluate 2k^×(3i^×2j^)2\hat{k}\times(3\hat{i}\times2\hat{j})

2k^×(3i^×2j^​)

Practice: Cross Product

Practice: Cross Product
Recall that i⃗=[1, 0, 0]\vec{i}=\left[1,\ 0,\ 0\right]i=[1, 0, 0], j⃗=[0, 1, 0]\vec{j}=\left[0,\ 1,\ 0\right]j​=[0, 1, 0], and k⃗=[0, 0, 1]\vec{k}=\left[0,\ 0,\ 1\right]k=[0, 0, 1].
Match the following cross products with the correct result.

Cross Product

Practice: Orthogonal Vectors
Find a vector orthogonal to both u⃗=[2−13]\vec u 
=
\begin{bmatrix}
  2\\ -1\\ 3
\end{bmatrix}u=​2−13​​ and v⃗=[0−1−1]\vec{v} = 
\begin{bmatrix}
  0\\ -1\\ -1\\
\end{bmatrix}v=​0−1−1​​. 
For extra practice, check your answer!

Practice Question: Orthogonal Vectors

Practice Question: Orthogonal Vectors
Find a vector orthogonal to both 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).v=(0,−1,−1).

Practice: Orthogonal Vectors

Practice: Orthogonal Vectors
Find a vector orthogonal to both 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).v=(0,−1,−1).

Practice: Cross Product

Practice: Cross Product
Consider u⃗=[−2, 3, −1]\vec{u}=\left[-2,\ 3,\ -1\right]u=[−2, 3, −1] and v⃗=[1, 0, −1]\vec{v}=\left[1,\ 0,\ -1\right]v=[1, 0, −1].

If u⃗=(2,0,−3), v⃗=(−2,2,−4)\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right)u=(2,0,−3), v=(−2,2,−4) and w⃗=(1,2,1)\vec{w}=\left(1,2,1\right)w=(1,2,1), find a vector that is orthogonal to uv⃗\vec{uv}uv and uw⃗\vec{uw}uw.

If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4),u⃗=(2,0,−3),v=(−2,2,−4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1)w⃗=(1,2,1), compute u⃗ × w⃗\vec{u}\ \times\ \vec{w}u × w.

Give that 𝑢⃗𝑢⃗u⃗ and 𝑣⃗\vec𝑣v are vectors in R3\mathbb{R}^3R3, which of the following statements is/are always true?
i. 2𝑢⃗+3𝑣⃗=2𝑣⃗+3𝑢⃗2𝑢⃗ + 3\vec𝑣 = 2\vec𝑣 + 3𝑢⃗2u⃗+3v=2v+3u⃗
ii. 2𝑢⃗.(−3𝑣⃗)=−2(3𝑢⃗.𝑣⃗)2𝑢⃗ . (−3\vec𝑣 ) = −2(3𝑢⃗ . \vec𝑣 )2u⃗.(−3v)=−2(3u⃗.v)

If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4),u⃗=(2,0,−3),v=(−2,2,−4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1)w⃗=(1,2,1), compute u⃗ × w⃗\vec{u}\ \times\ \vec{w}u × w.

If u=(2,0,−3), v=(−2,2,−4)u=\left(2,0,-3\right),\ v=\left(-2,2,-4\right)u=(2,0,−3), v=(−2,2,−4) and w=(1,2,1)w=\left(1,2,1\right)w=(1,2,1), find a vector that is orthogonal to uv⃗\vec{uv}uv and uw⃗\vec{uw}uw.

Dot and Cross Products: Vector Norm

If u⃗, v⃗, w⃗  ∈ R3\vec{u},\ \vec{v},\ \vec{w}\ \ \in\ R^3u, v, w  ∈ R3, which of the following operations is/are defined?
(i.e. which of the following can be calculated?)
Select all that are correct.

Practice Question: Vector Product

Related Topics

Dot Product

Cross Product

Practice Question: Vector Product

More Dot Product Questions:

133 - FML 3 - 18.1W e.g. 20

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 40_$\tkcth{Mock F}\tkct{?}$_$\tkct{no soln / vid}$

Example: Orthogonal Vectors

Practice Question: Orthogonal Vectors

Practice Question: Orthogonal Vectors

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 52

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock ✓}$ 133 - FML 3 - 18.1W e.g. 46_$\tkcth{Mock F}\tkct{?}$_$\tkct{vid soln only}$

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 54

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ | Basic dot product.

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |

Dot and Cross Products: Vector Norm

Unit Vectors: Dot Product

More Cross Product Questions:

Cross Product

Find vec form line thru pt ortho to 2 vec form lines

Practice Question: Orthogonal Vectors

Practice: Cross Product

Cross Product

Practice Question: Orthogonal Vectors

Practice: Orthogonal Vectors

Practice: Cross Product

Dot and Cross Products: Vector Norm