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Dot and Cross Products: Vector Norm

Related Topics

Wize University Linear Algebra Textbook > Vectors

Length of a Vector (Vector Norm)

5 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Cross Product

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Dot Product

3 Activities

If u⃗, v⃗, w⃗  ∈ 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.
More Length of a Vector (Vector Norm) Questions:
Write the vectors𝑢⃗=(1,2,3)𝑢⃗=(1,2,3), 𝑣⃗=(5,−2,2)𝑣⃗=(\sqrt{5},−2,2), and  𝑤⃗=(3,3,3)\ 𝑤⃗=(\sqrt{3},\sqrt{3},3) in increasing order of length.
(i.e. shortest vector first)
Vectors and coordinates (Components of a vector)
In diagram below two vectors A⃗\vec{A}and B⃗\vec{B} have shown. VectorsC⃗\vec{C}and D⃗\vec{D}are the sum (A⃗+B⃗)\left(\vec{A}+\vec{B}\right) and difference (A⃗−B⃗)\left(\vec{A}-\vec{B}\right)vectors, respectively.
Which statement is correct about the magnitude of four vectors?
Practice Question: Properties of Vector Operations
Practice Question: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, which of the following statements is/are always true?
i.) ∣∣v⃗−u⃗∣∣=−∣∣u⃗−v⃗∣∣\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
Practice: Magnitude of a 3D Vector
Practice: Magnitude of a 3D Vector
An aircraft's location is 200 miles S30oW from base, and it's altitude is 35,000 ft. Find the distance between the aircraft and the base.
$\tkct{cut from 19.4F}$ 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} and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}, where u⃗⋅v⃗=2\bcb{\vec{u} \cdot \vec{v} = 2}, find ∣u⃗−v⃗∣\bcb{|{\vec{u} - \vec{v} }|}.
133 - FML 3 - 18.1W e.g. 25
Given the vector v⃗=<v1,v2,v3>\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,} and v3\bcb{v_3}.
Length / magnitude / norm. 19.1W
1. Find the vector d⃗\vd connecting the points P1(−2,−1)P_1(-2,-1) and P2(3,11)P_2(3,11).
2. Find the length of the vector d⃗\vd found, above. What does this length represent?
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
133 - FML 3 - 18.1W e.g. 25
Given the vector v⃗=<v1,v2,v3>\bcb{\vec{v} = \left<v_1, v_2, v_3 \right>}, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,} and v3\bcb{v_3}.
Practice Question: Properties of Vector Operations
Practice Question: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, which of the following statements is/are always true?
i.) ∣∣v⃗−u⃗∣∣=−∣∣u⃗−v⃗∣∣\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
Practice Question: Vector Operations and Length
Practice Question: Properties of Vector Operations
If u⃗, v⃗, w⃗ ∈R3\vec{u},\ \vec{v},\ \vec{w}\ \in R^3, which of the following statements is/are always true?
i.) ∣∣v⃗−u⃗∣∣=−∣∣u⃗−v⃗∣∣\left|\left|\vec{v}-\vec{u}\right|\right|=-\left|\left|\vec{u}-\vec{v}\right|\right|
Practice: Norm of a Vector
Which of the following statements are true:
There exist vectors such that ∥u⃗+v⃗∥=∥u⃗∥+∥v⃗∥\|\vec{u}+\vec{v}\|=\|\vec{u}\|+\|\vec{v}\|
If ∥u⃗∥=∥v⃗∥\|\vec{u}\|=\|\vec{v}\| then u⃗=v⃗\vec{u}=\vec{v}
Vector Norm
Practice: Properties of the Norm
Consider u⃗, v⃗, w⃗ ∈Rn\vec{u},\ \vec{v},\ \vec{w}\ \in \reals^n. Select all of the statements that are always true.
Vector Norm
Practice: Distance Between Two Points/Vectors
Vector Norm
Practice: Vector Length
Given v⃗=[−3k]\vec{v}= \begin{bmatrix} -3\\ k\\ \end{bmatrix} and ∥v⃗∥=5\lVert \vec v \rVert = 5 , find all possible values of kk.
Practice Question: Vector Length
Practice Question: Vector Length
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
Length of a Vector

Practice: Vector Length
Practice: Vector Length
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘) is 5, find all possible values of 𝑘.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |
The distance between the points P1(2,−3,7)P_1(2,-3,7) and P2(4,3,16)P_2(4,3,16) is:
Vector Norm: Length of a Vector
Find ∥v∥\lVert \bm{v} \rVert where v=(2,3,−3,−4,0)\bm{v}=(\sqrt 2,3,-3,-4,0)
More Cross Product Questions:
Cross Product
Practice: Orthogonal Vectors
Find a vector orthogonal to both u⃗=[2−13]\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix} and v⃗=[0−1−1]\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,−3)\left(1,2,-3\right) 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) and (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] and [−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) and 𝑣⃗=(0,−1,−1).\vec{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}, and C⃗=−2i^+j^\vec{C}=-2\hat{i}+\hat{j}. The angle between C⃗\vec{C} and A⃗×B⃗\vec{A}\times\vec{B} is?
Evaluate 2k^×(3i^×2j^)2\hat{k}\times(3\hat{i}\times2\hat{j})
Practice: Cross Product
Practice: Cross Product
Recall that i⃗=[1, 0, 0]\vec{i}=\left[1,\ 0,\ 0\right], j⃗=[0, 1, 0]\vec{j}=\left[0,\ 1,\ 0\right], and k⃗=[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 u⃗=[2−13]\vec u = \begin{bmatrix} 2\\ -1\\ 3 \end{bmatrix} and v⃗=[0−1−1]\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 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).
Practice: Orthogonal Vectors
Practice: Orthogonal Vectors
Find a vector orthogonal to both 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).
Practice: Cross Product
Practice: Cross Product
Consider u⃗=[−2, 3, −1]\vec{u}=\left[-2,\ 3,\ -1\right] and v⃗=[1, 0, −1]\vec{v}=\left[1,\ 0,\ -1\right].
If u⃗=(2,0,−3), v⃗=(−2,2,−4)\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right) and w⃗=(1,2,1)\vec{w}=\left(1,2,1\right), find a vector that is orthogonal to uv⃗\vec{uv} and uw⃗\vec{uw}.
If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1), compute u⃗ × w⃗\vec{u}\ \times\ \vec{w}.
Give that 𝑢⃗𝑢⃗ and 𝑣⃗\vec𝑣 are vectors in R3\mathbb{R}^3, which of the following statements is/are always true?
i. 2𝑢⃗+3𝑣⃗=2𝑣⃗+3𝑢⃗2𝑢⃗ + 3\vec𝑣 = 2\vec𝑣 + 3𝑢⃗
ii. 2𝑢⃗.(−3𝑣⃗)=−2(3𝑢⃗.𝑣⃗)2𝑢⃗ . (−3\vec𝑣 ) = −2(3𝑢⃗ . \vec𝑣 )
If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1), compute u⃗ × w⃗\vec{u}\ \times\ \vec{w}.
If u=(2,0,−3), v=(−2,2,−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 uv⃗\vec{uv} and uw⃗\vec{uw}.
Practice Question: Vector Product
Practice Question: Vector Product
Given that 𝑢⃗=(1,1,−1)𝑢⃗=(1,1,−1) and 𝑣⃗=(0,−2,2),𝑣⃗=(0,−2,2), compute (𝑢⃗⋅𝑣⃗)(𝑣⃗×𝑢⃗).(𝑢⃗\cdot𝑣⃗)(𝑣⃗\times𝑢⃗).
More Dot Product Questions:
133 - FML 3 - 18.1W e.g. 20
For two vectors v⃗\bcb{\vv} and w⃗\bcb{\vw} ∈  I ⁣Rn\bcb{\in}\; \R{n}, v⃗⋅w⃗:=‾\bcb{\vv \cdot \vw :=\underline{\qquad\qquad\qquad}}.
$\tkct{cut from 19.4F}$ 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>} and v⃗=<2, −1, 3>\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}, find (2u⃗)⋅(3v⃗)\bcb{(2\vec{u})\cdot (3\vec{v})}.
Example: Orthogonal Vectors
Example:
Find all values of kk such that the vectorsv⃗=(k, 1)\vec{v}=\left(k,\ 1\right) and u⃗=(k,−1)\vec{u}=\left(k,-1\right) are orthogonal.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find the value(s) of cc such that u⃗=(−1,c,2) \vec{u}=(-1,c,2)\ and v⃗=(−7, c, −4)\vec{v}=\left(-7,\ c,\ -4\right) are orthogonal.
Practice Question: Orthogonal Vectors
Practice Question: Orthogonal Vectors
Find the value(s) of cc such that u⃗=(−1,c,2) \vec{u}=(-1,c,2)\ and v⃗=(−7, c, −4)\vec{v}=\left(-7,\ c,\ -4\right) 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} and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3} and u⃗\bcb{\vec{u}} has the same direction as v⃗\bcb{\vec{v}} then (3u⃗)⋅(2v⃗)\bcb{(3\vec{u}) \cdot (2\vec{v})} 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} and ∣v⃗∣=3\bcb{|{ \vec{v} |} = 3}, where u⃗⋅v⃗=2\bcb{\vec{u} \cdot \vec{v} = 2}, find ∣u⃗−v⃗∣\bcb{|{\vec{u} - \vec{v} }|}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 54
Find the value of k\bcb{k} such that the vector u⃗=<1,k,3>\bcb{\vec{u} = \left< 1, k, 3 \right>} is perpendicular to the vector v⃗=<2,−1,k>\bcb{\vec{v} = \left< 2, -1, k\right>}.
If 𝑢⃗=(2,0,−3),𝑣⃗=(−2,2,−4),𝑢⃗ = (2, 0, −3), \vec𝑣 = (−2, 2, −4), and 𝑤⃗=(1,2,1)𝑤⃗= (1,2,1), compute u⃗ ∙ v⃗\vec{u}\ \bullet\ \vec{v}.
Practice Question: Vector Product
Practice Question: Vector Product
Given that 𝑢⃗=(1,1,−1)𝑢⃗=(1,1,−1) and 𝑣⃗=(0,−2,2),𝑣⃗=(0,−2,2), compute (𝑢⃗⋅𝑣⃗)(𝑣⃗×𝑢⃗).(𝑢⃗\cdot𝑣⃗)(𝑣⃗\times𝑢⃗).
Find the value(s) of kk(if any) such that the vectors u→=(1,k,1,0,−1)\overrightarrow{u}=\left(1,k,1,0,-1\right)and v→=(k,2,0,3,−4)\overrightarrow{v}=\left(k,2,0,3,-4\right)are orthogonal.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ | Basic dot product.
The dot product v⃗⋅w⃗\vv \cdot \vw, for v⃗=<−1, 1, −1>\vv = \rowvecth{-1}{1}{-1} and w⃗=<−2, −2, 3>\vw = \rowvecth{-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} 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) and P2(1,−1,1)P_2(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) and P2(−2,0,1)P_2(-2,0,1) is:
Unit Vectors: Dot Product
Consider the vectors u=i−4k\bm{u}=\bm{i}-4\bm{k} and v=3i−2j+k\bm{v}=3\bm{i}-2\bm{j}+\bm{k} in R3\reals^3. Find (if possible) 3v⋅(k−2u)3\bm{v}\cdot(\bm{k}-2\bm{u}).