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

Unit Vectors

3 Activities

Wize University Linear Algebra Textbook > Products of Vectors

Projection (Proj) and Perpendicular (Perp)

6 Activities

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 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⟩.

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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. 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⟩

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 Projection (Proj) and Perpendicular (Perp) Questions:

Example: Dot Product (2)

Find the scalar and vector projection of b⃗=⟨2,1,3⟩\vec{b}=\langle 2,1,3\rangleb=⟨2,1,3⟩onto a⃗=⟨−1,4,2⟩\vec{a}=\langle -1,4,2\ranglea=⟨−1,4,2⟩.

Find the vector projection of  [3,2][3,2][3,2] onto the vector [3,6]\left[3,6\right][3,6]

133 - FML 3 - 18.1W - e.g. 9

Find the vector-component ı^∣∣d⃗ \ihat_{||_{\vd}} \,^∣∣d​​ of ı^\bcb{\boldsymbol{ \ihat}}^ in the direction of the line given by y=mx\bcb{\boldsymbol{ y=mx}}y=mx. Repeat for ȷ^∣∣d⃗ \jhat_{||{\vd}} \, ^​∣∣d​, the vector-component of ȷ^\bcb{\boldsymbol{ \jhat }}^​ in the direction of the line.

133 - FML 3 - 18.1W e.g. 58.3

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the point on the line L\bcb{L}L closest to the point P2\bcb{P_2}P2​.

133 - FML 3 - 18.1W e.g. 58.2

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0)
Find the minimum distance of P2\bcb{P_2}P2​ to the line L\bcb{L}L found above. 

133 - FML 3 - 18.1W e.g. 43

 Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>}u=⟨8,2,2⟩ and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}v=⟨2,−1,−1⟩, find the vector projection of u⃗\bcb{\vec{u}}u onto v⃗\bcb{\vec{v}}v. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

Let P=(−3,−3,−4)\bcb{P = (-3, -3, -4)}P=(−3,−3,−4), Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}Q=(−5,−6,−3), R=(−6,−4,−6)\bcb{R = (-6,-4,-6)}R=(−6,−4,−6) and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}S=(−2,−3,−4). Let l1\bcb{l_1}l1​ be the line passing through P\bcb{P}P and Q\bcb{Q}Q, and let l2\bcb{l_2}l2​ be the line passing through R\bcb{R}R and S\bcb{S}S. Find the distance between R\bcb{R}R and l1\bcb{l_1}l1​. 

133 - FML 3 - 18.1W e.g. 36

For two lines having directional vectors v1\mathbf{v}_1v1​ and v2\mathbf{v}_2v2​, respectively, the shortest distance between them can be found by creating a vector that connects the two lines (by subtracting a point on one line from a point on the other), i.e. (r2−r1)\left(\mathbf{r}_2 - \mathbf{r}_1\right)(r2​−r1​), and projecting it onto the unit normal vector connecting the two lines. The unit normal is easily found by crossing their respective directionals and dividing by the magnitude of the resulting vector, as illustrated in Fig. 5. i.e.
s=∣projn^(r2−r1)∣=∣(r2−r1)⋅v1×v2∣v1×v2∣∣
 	s = \left| \text{proj}_{\mathbf{\hat{n}}} \left(\mathbf{r}_2 - \mathbf{r}_1\right)\right| = \left| \left(\mathbf{r}_2 - \mathbf{r}_1\right) \cdot \frac{\mathbf{v}_1 \times \mathbf{v}_2}{\left|\mathbf{v}_1 \times \mathbf{v}_2\right|}\right|
 s=∣projn^​(r2​−r1​)∣=​(r2​−r1​)⋅∣v1​×v2​∣v1​×v2​​​
Find the equations of the line passing through the point (2,3,−4)\bcb{(2, 3, -4)}(2,3,−4) parallel to the vector i−4k\bcb{\mathbf{i} - 4\mathbf{k}}i−4k ; express it  in vector-parametric, scalar-parametric and standard form. 
 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.3

 Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)}P1​=(0,0,1) and  P2=(1,−2,3)\bcb{P_2 = (1,-2,3)}P2​=(1,−2,3) and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}v=⟨2,−3,1⟩, find the point Po\bcb{P_o}Po​ on the plane Π: 2x−3y+z=1\bcb{\Pi: \ 2x - 3y + z = 1}Π: 2x−3y+z=1 that is closest to the point P2\bcb{P_2}P2​.

$\tkct{cut from 19.4F}$ 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⟩.

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 43

 Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>}u=⟨8,2,2⟩ and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}v=⟨2,−1,−1⟩, find the vector projection of u⃗\bcb{\vec{u}}u onto v⃗\bcb{\vec{v}}v. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0), find the minimum distance of P2\bcb{P_2}P2​ to the line L\bcb{L}L that is parallel to v⃗\bcb{\vec{v}}v that passes through P1\bcb{P_1}P1​. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.2

Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)}P1​=(0,0,1) and  P2=(1,−2,3)\bcb{P_2 = (1,-2,3)}P2​=(1,−2,3) and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}v=⟨2,−3,1⟩
Find the minimum distance between the point P2\bcb{P_2}P2​ and the plane Π: 2x−3y+z=1\bcb{\Pi:\ 2x-3y + z = 1}Π: 2x−3y+z=1. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.3

Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>}v=⟨1,1,2⟩ and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)}P1​=(0,−1,0) and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}P2​=(−1,1,0), find the point on 
the line L\bcb{L}L (which is parallel to v⃗\bcb{\vec{v}}v that passes through P1\bcb{P_1}P1​) that is closest to the point P2\bcb{P_2}P2​.

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

Related Topics

Unit Vectors

Projection (Proj) and Perpendicular (Perp)

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⟩.

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. 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

Unit Vectors: Dot Product

More Projection (Proj) and Perpendicular (Perp) Questions:

Example: Dot Product (2)

133 - FML 3 - 18.1W - e.g. 9

133 - FML 3 - 18.1W e.g. 58.3

133 - FML 3 - 18.1W e.g. 58.2

133 - FML 3 - 18.1W e.g. 43

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 41

133 - FML 3 - 18.1W e.g. 36

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.3

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

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 43

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.2

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.2

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.3

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