$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock ✓}$ 133 - FM…

Dot Product

3 Activities

Wize University Linear Algebra Textbook > Vectors

Length of a Vector (Vector Norm)

5 Activities

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∣.

|\vec{u} - \vec{v} |=

I don't know

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 $\tkco{ S}$ | 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.

Practice Question: Vector Product

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).

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).

More Length of a Vector (Vector Norm) Questions:

Write the vectors𝑢⃗=(1,2,3)𝑢⃗=(1,2,3)u⃗=(1,2,3), 𝑣⃗=(5,−2,2)𝑣⃗=(\sqrt{5},−2,2)v⃗=(5​,−2,2), and  𝑤⃗=(3,3,3)\ 𝑤⃗=(\sqrt{3},\sqrt{3},3) w⃗=(3​,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}Aand B⃗\vec{B}B have shown. VectorsC⃗\vec{C}Cand D⃗\vec{D}Dare the sum (A⃗+B⃗)\left(\vec{A}+\vec{B}\right)(A+B) and difference (A⃗−B⃗)\left(\vec{A}-\vec{B}\right)(A−B)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^3u, v, w ∈R3, 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|∣∣v−u∣∣=−∣∣u−v∣∣

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.

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>}v=⟨v1​,v2​,v3​⟩, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,}v1​,v2​, and v3\bcb{v_3}v3​.

Length / magnitude / norm. 19.1W

1. Find the vector d⃗\vdd connecting the points P1(−2,−1)P_1(-2,-1)P1​(−2,−1) and P2(3,11)P_2(3,11)P2​(3,11).
2. Find the length of the vector d⃗\vdd found, above. What does this length represent? 

Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘)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>}v=⟨v1​,v2​,v3​⟩, find an expression for its length in terms of its components v1, v2,\bcb{v_1,\, v_2,}v1​,v2​, and v3\bcb{v_3}v3​.

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^3u, v, w ∈R3, 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|∣∣v−u∣∣=−∣∣u−v∣∣

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^3u, v, w ∈R3, 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|∣∣v−u∣∣=−∣∣u−v∣∣

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}\|∥u+v∥=∥u∥+∥v∥
If ∥u⃗∥=∥v⃗∥\|\vec{u}\|=\|\vec{v}\|∥u∥=∥v∥ then u⃗=v⃗\vec{u}=\vec{v}u=v

Vector Norm

Practice: Properties of the Norm
Consider u⃗, v⃗, w⃗ ∈Rn\vec{u},\ \vec{v},\ \vec{w}\ \in \reals^nu, v, w ∈Rn. 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}v=[−3k​] and  ∥v⃗∥=5\lVert \vec v \rVert = 5∥v∥=5 , find all possible values of kkk.

Practice Question: Vector Length

Practice Question: Vector Length
Given that the length of 𝑣⃗=(−3,𝑘)\vec{𝑣}=(−3,𝑘)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)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)P1​(2,−3,7) and P2(4,3,16)P_2(4,3,16)P2​(4,3,16) 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.

Vector Norm: Length of a Vector

Find ∥v∥\lVert \bm{v} \rVert∥v∥ where v=(2,3,−3,−4,0)\bm{v}=(\sqrt 2,3,-3,-4,0)v=(2​,3,−3,−4,0)

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock ✓}$ 133 - FM…

Related Topics

Dot Product

Length of a Vector (Vector Norm)

If $\bcb{|{\vec{u}|} = 2}$ and $\bcb{|{ \vec{v} |} = 3}$ , where $\bcb{\vec{u} \cdot \vec{v} = 2}$ , find $\bcb{|{\vec{u} - \vec{v} }|}$ .

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 $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 54

Practice Question: Vector Product

$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ | 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 Length of a Vector (Vector Norm) Questions:

Vectors and coordinates (Components of a vector)

Practice Question: Properties of Vector Operations

Practice: Magnitude of a 3D Vector

133 - FML 3 - 18.1W e.g. 25

Length / magnitude / norm. 19.1W

133 - FML 3 - 18.1W e.g. 25

Practice Question: Properties of Vector Operations

Practice Question: Vector Operations and Length

Practice: Norm of a Vector

Vector Norm

Vector Norm

Vector Norm

Practice Question: Vector Length

Length of a Vector

Practice: Vector Length

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

Dot and Cross Products: Vector Norm

Vector Norm: Length of a Vector