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

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Wize University Linear Algebra Textbook > Vectors

Unit Vectors

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Dot Product

3 Activities

A unit vector perpendicular to the vector wβƒ—=<2, 5>\vw = \rowvec{2}{5} is:
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)
Practice: Cross Product
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].
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:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
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 Dot Product Questions:
133 - FML 3 - 18.1W e.g. 20
For two vectors vβƒ—\bcb{\vv} and wβƒ—\bcb{\vw} βˆˆβ€…β€ŠI ⁣Rn\bcb{\in}\; \R{n}, vβƒ—β‹…wβƒ—:=β€Ύ\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 uβƒ—=<1,βˆ’2,3>\bcb{\vec{u} = \left< 1, -2, 3 \right>} and vβƒ—=<2,β€‰βˆ’1, 3>\bcb{\vec{v} = \left< 2,\, -1,\, 3 \right>}, find (2uβƒ—)β‹…(3vβƒ—)\bcb{(2\vec{u})\cdot (3\vec{v})}.
Example: Orthogonal Vectors
Example:
Find all values of kk such that the vectorsvβƒ—=(k, 1)\vec{v}=\left(k,\ 1\right) and uβƒ—=(k,βˆ’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 uβƒ—=(βˆ’1,c,2) \vec{u}=(-1,c,2)\ and vβƒ—=(βˆ’7, c, βˆ’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 uβƒ—=(βˆ’1,c,2) \vec{u}=(-1,c,2)\ and vβƒ—=(βˆ’7, c, βˆ’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 ∣uβƒ—βˆ£=2\bcb{|{ \vec{u}| } = 2} and ∣vβƒ—βˆ£=3\bcb{|{ \vec{v} |} = 3} and uβƒ—\bcb{\vec{u}} has the same direction as vβƒ—\bcb{\vec{v}} then (3uβƒ—)β‹…(2vβƒ—)\bcb{(3\vec{u}) \cdot (2\vec{v})} is:
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | $\tkco{duplicate to mock βœ“}$ 133 - FML 3 - 18.1W e.g. 46_$\tkcth{Mock F}\tkct{?}$_$\tkct{vid soln only}$
If ∣uβƒ—βˆ£=2\bcb{|{\vec{u}|} = 2} and ∣vβƒ—βˆ£=3\bcb{|{ \vec{v} |} = 3}, where uβƒ—β‹…vβƒ—=2\bcb{\vec{u} \cdot \vec{v} = 2}, find ∣uβƒ—βˆ’vβƒ—βˆ£\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 uβƒ—=<1,k,3>\bcb{\vec{u} = \left< 1, k, 3 \right>} is perpendicular to the vector vβƒ—=<2,βˆ’1,k>\bcb{\vec{v} = \left< 2, -1, k\right>}.
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— βˆ™ vβƒ—\vec{u}\ \bullet\ \vec{v}.
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).
Find the value(s) of kk(if any) such that the vectors uβ†’=(1,k,1,0,βˆ’1)\overrightarrow{u}=\left(1,k,1,0,-1\right)and vβ†’=(k,2,0,3,βˆ’4)\overrightarrow{v}=\left(k,2,0,3,-4\right)are orthogonal.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ | Basic dot product.
The dot product vβƒ—β‹…wβƒ—\vv \cdot \vw, for vβƒ—=<βˆ’1, 1,β€‰βˆ’1>\vv = \rowvecth{-1}{1}{-1} and wβƒ—=<βˆ’2,β€‰βˆ’2, 3>\vw = \rowvecth{-2}{-2}{3}, is equal to:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
A vector perpendicular to the vector connecting the points P1(2,2,2)P_1(2,2,2) and P2(1,βˆ’1,1)P_2(1,-1,1) is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz βœ“}$ |
A vector perpendicular to the vector connecting the points P1(0,1,βˆ’2)P_1(0,1,-2) and P2(βˆ’2,0,1)P_2(-2,0,1) is:
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.
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}).