Unit Vectors: Dot Product

Unit Vectors

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Dot Product

3 Activities

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).

Vectors in 2D and 3D Systems (2)

Practice
Find the unit vector that points in the same direction as a⃗ =<−10, 4, −3>\vec{a}~=<-10,~4,~-3>a =<−10, 4, −3>.

Unit Vectors

Practice: Unit Vectors
Find the value of kkk such that  u⃗=⟨k,12,−k⟩\vec u = \left\lang k, \dfrac{1}{\sqrt2}, -k \right\rangu=⟨k,2​1​,−k⟩  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>}v=⟨v1​,v2​,v3​⟩.

Practice: Unit Vector

Practice: Unit Vectors
Find the unit vector in the same direction as v⃗=(1,−2,−1)\vec v = (1,-2,-1)v=(1,−2,−1).

Practice: Unit Vector

Practice: Unit Vector
Find the unit vector in the same direction as i⃗−5k⃗\vec i-5\vec ki−5k.

Practice: Unit Vector

Practice: Unit Vector
Find the unit vector in the same direction as i^−5k^\hat{i}-5\hat{k}i^−5k^.

Concept Clarifier

Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)v=(3,4,5)

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].

133 - FML 3 - 18.1W e.g. 24

If a vector v⃗\bcb{\vec{v}}v has length L\bcb{\mathcal{L}}L, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}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>}v=⟨v1​,v2​,v3​⟩.

133 - FML 3 - 18.1W e.g. 48

 Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>}u=⟨102,112,−56⟩ onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}v=⟨0,3,4⟩.

133 - FML 3 - 18.1W e.g. 24

If a vector v⃗\bcb{\vec{v}}v has length L\bcb{\mathcal{L}}L, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}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 u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>}u=⟨102,112,−56⟩ onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}v=⟨0,3,4⟩.

Concept Clarifier

Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)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)v=(3,4,5)

Practice Question: Unit Vector

Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −1).

Practice Question: Unit Vector

Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −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,−1,0,−1,0)𝑣⃗\ =(1,−1,0,−1,0)v⃗ =(1,−1,0,−1,0).

Practice: Unit Vector

Practice: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −1).

Practice: Unit Vector

Practice: Unit Vector
Find the unit vector in the same direction as i⃗−5k⃗\vec i-5\vec ki−5k.

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

A unit vector parallel to the vector v⃗=<4, −3>\vv = \rowvec{4}{-3}v=⟨4,−3⟩ is: 

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

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

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

Find a vector parallel to the vector u⃗=<1, 4, 8>\vu = \rowvecth{1}{4}{8}u=⟨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 u⃗=<2, 1, −2>\vu = \rowvecth{2}{1}{-2}u=⟨2,1,−2⟩

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.

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

Related Topics

Unit Vectors

Dot Product

More Unit Vectors Questions:

Vectors in 2D and 3D Systems (2)

Unit Vectors

133 - FML 3 - 18.1W e.g. 26

Practice: Unit Vector

Practice: Unit Vector

Practice: Unit Vector

Concept Clarifier

Practice: Cross Product

133 - FML 3 - 18.1W e.g. 24

133 - FML 3 - 18.1W e.g. 26

133 - FML 3 - 18.1W e.g. 48

133 - FML 3 - 18.1W e.g. 24

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 48

Concept Clarifier

Unit Vector Practice Question

Practice Question: Unit Vector

Practice Question: Unit Vector

Practice Question: Unit Vector

Practice Question: Unit Vector

Practice: Unit Vector

Practice: Unit Vector

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

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

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

DIY Unit Vector

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

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