Find vec form line thru pt ortho to 2 vec form lines

Cross Product

3 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Lines in $\reals^n$

6 Activities

Find the equation of the line that passes through the point (1,2,−3)\left(1,2,-3\right)(1,2,−3) 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)(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)(x,y,z)=(3,−1,6)+t(0,2,1)

(x,y,z)=(0,0,−1)+t(1,2,−3)(x,y,z)=\left(0,0,-1\right)+t\left(1,2,-3\right)(x,y,z)=(0,0,−1)+t(1,2,−3)

(x,y,z)=(1,2,−3)+t(0,0,−1)(x,y,z)=\left(1,2,-3\right)+t\left(0,0,-1\right)(x,y,z)=(1,2,−3)+t(0,0,−1)

(x,y,z)=(1,2,−3)+t(2,2,−1)(x,y,z)=\left(1,2,-3\right)+t\left(2,2,-1\right)(x,y,z)=(1,2,−3)+t(2,2,−1)

(x,y,z)=(2,−1,2)+t(1,2,−3)(x,y,z)=\left(2,-1,2\right)+t\left(1,2,-3\right)(x,y,z)=(2,−1,2)+t(1,2,−3)

(x,y,z)=(1,2,−3)+t(2,−1,2)(x,y,z)=\left(1,2,-3\right)+t\left(2,-1,2\right)(x,y,z)=(1,2,−3)+t(2,−1,2)

I don't know

Cross Product

Practice: Orthogonal Vectors
Find a vector orthogonal to both u⃗=[2−13]\vec u 
=
\begin{bmatrix}
  2\\ -1\\ 3
\end{bmatrix}u=​2−13​​ and v⃗=[0−1−1]\vec{v} = 
\begin{bmatrix}
  0\\ -1\\ -1\\
\end{bmatrix}v=​0−1−1​​. 
For extra practice, check your answer!

Which of the following vectors is orthogonal to both [1,2,3][1,2,3][1,2,3] and [−3,1,0][-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)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).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}A=2i^+j^​,B=i^−2j^​, and C⃗=−2i^+j^\vec{C}=-2\hat{i}+\hat{j}C=−2i^+j^​. The angle between C⃗\vec{C}C and A⃗×B⃗\vec{A}\times\vec{B}A×B is?

Evaluate 2k^×(3i^×2j^)2\hat{k}\times(3\hat{i}\times2\hat{j})

2k^×(3i^×2j^​)

Practice: Cross Product

Practice: Cross Product
Recall that i⃗=[1, 0, 0]\vec{i}=\left[1,\ 0,\ 0\right]i=[1, 0, 0], j⃗=[0, 1, 0]\vec{j}=\left[0,\ 1,\ 0\right]j​=[0, 1, 0], and k⃗=[0, 0, 1]\vec{k}=\left[0,\ 0,\ 1\right]k=[0, 0, 1].
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}u=​2−13​​ and v⃗=[0−1−1]\vec{v} = 
\begin{bmatrix}
  0\\ -1\\ -1\\
\end{bmatrix}v=​0−1−1​​. 
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)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).v=(0,−1,−1).

Practice: Orthogonal Vectors

Practice: Orthogonal Vectors
Find a vector orthogonal to both 𝑢⃗=(2,−1,3)𝑢⃗=(2,−1,3)u⃗=(2,−1,3) and 𝑣⃗=(0,−1,−1).\vec{𝑣} = (0, −1, −1).v=(0,−1,−1).

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].

If u⃗=(2,0,−3), v⃗=(−2,2,−4)\vec{u}=\left(2,0,-3\right),\ \vec{v}=\left(-2,2,-4\right)u=(2,0,−3), v=(−2,2,−4) and w⃗=(1,2,1)\vec{w}=\left(1,2,1\right)w=(1,2,1), find a vector that is orthogonal to uv⃗\vec{uv}uv and uw⃗\vec{uw}uw.

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 × w⃗\vec{u}\ \times\ \vec{w}u × w.

Give that 𝑢⃗𝑢⃗u⃗ and 𝑣⃗\vec𝑣v are vectors in R3\mathbb{R}^3R3, which of the following statements is/are always true?
i. 2𝑢⃗+3𝑣⃗=2𝑣⃗+3𝑢⃗2𝑢⃗ + 3\vec𝑣 = 2\vec𝑣 + 3𝑢⃗2u⃗+3v=2v+3u⃗
ii. 2𝑢⃗.(−3𝑣⃗)=−2(3𝑢⃗.𝑣⃗)2𝑢⃗ . (−3\vec𝑣 ) = −2(3𝑢⃗ . \vec𝑣 )2u⃗.(−3v)=−2(3u⃗.v)

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 × w⃗\vec{u}\ \times\ \vec{w}u × w.

If u=(2,0,−3), v=(−2,2,−4)u=\left(2,0,-3\right),\ v=\left(-2,2,-4\right)u=(2,0,−3), v=(−2,2,−4) and w=(1,2,1)w=\left(1,2,1\right)w=(1,2,1), find a vector that is orthogonal to uv⃗\vec{uv}uv and uw⃗\vec{uw}uw.

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).

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.

More Lines in $\reals^n$ Questions:

Consider u=(1,2,3) and v=(2,-1,4)
a. Find a parametric equation of the line passing through the point (1,3,1) and 
    perpendicular to both u and v.

1) Find a vector of length 1 that is parallel to the line given by the equations:
𝑦−𝑧=42𝑥+𝑦+3𝑧=2\begin{array}{}𝑦−𝑧=4\\2𝑥+𝑦+3𝑧=2

\end{array}y−z=42x+y+3z=2​

a) Write the parametric equation of line 
L:{x+2y=0x−z=5L: \begin{cases}
x+2y=0\\
x-z=5
\end{cases}L:{x+2y=0x−z=5​
b) Find intersection point of the line L with the plane P:3y−3z=2.P:3y-3z=2.P:3y−3z=2.

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)(1,2,−3) and is orthogonal to both lines
l1: (−5, 0, 1)+t(1,0,−1)l_1:\ \left(-5,\ 0,\ 1\right)+t\left(1,0,-1\right)l1​: (−5, 0, 1)+t(1,0,−1) and l2: (3,−1,6)+t(0,2,1)l_2:\ \left(3,-1,6\right)+t\left(0,2,1\right)l2​: (3,−1,6)+t(0,2,1).

Practice Question: Two Point Form Equation
Write a vector that is parallel to the line (x, y, z)=(1−t)(6,−1,2)+t(9,5,−7)\left(x,\ y,\ z\right)=\left(1-t\right)\left(6,-1,2\right)+t\left(9,5,-7\right)(x, y, z)=(1−t)(6,−1,2)+t(9,5,−7).

Which of the following vectors is a normal vector of a line that is parallel to 2x+y=−32x+y=-32x+y=−3?

133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5

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 them in vector-parametric, scalar-parametric and standard form.

133 - FML 3 - 18.1W e.g. 50

 Find the equation of a plane perpendicular to w⃗=<1,−2,1>\bcb{\vec{w} = \left< 1, -2, 1\right>}w=⟨1,−2,1⟩ that contains the point P=(−1,1,2)\bcb{P = (-1, 1, 2)}P=(−1,1,2). 

Lines in $\reals^n$

Practice: Lines in Rn\colorOne{\reals^n}Rn
Various pieces of information can be given to find the equation of a 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. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56

Given the points P1=(1,0,5)\bcb{P_1 = (1, 0, 5)}P1​=(1,0,5) and P2=(3,1,−2)\bcb{P_2 = (3, 1, -2)}P2​=(3,1,−2), which of the following points are on the line connecting P1\bcb{P_1}P1​ and P2\bcb{P_2}P2​?

$\tkct{cut from 19.4F}$ 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​. 

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1

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 scalar-parametric equation of the line L\bcb{L}L  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. 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​. 

133 - FML 3 - 18.1W - e.g. 8

If a line is given by y=mx\bcb{\boldsymbol{ y = mx}}y=mx, find a unit vector pointing in the direction of the line (i.e. find the directional vector).

Equation of Lines

Which of of the following is an equation of the line that passes through A(0,−3)A(0,-3)A(0,−3) and B(−3,6)B(-3,6)B(−3,6).

Find vec form line thru pt ortho to 2 vec form lines

Related Topics

Cross Product

Lines in $\reals^n$

More Cross Product Questions:

Cross Product

Practice Question: Orthogonal Vectors

Practice: Cross Product

Cross Product

Practice Question: Orthogonal Vectors

Practice: Orthogonal Vectors

Practice: Cross Product

Practice Question: Vector Product

Dot and Cross Products: Vector Norm

More Lines in $\reals^n$ Questions:

Find vec form line thru pt ortho to 2 vec form lines

133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5

133 - FML 3 - 18.1W e.g. 50

Lines in $\reals^n$

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

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

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56

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

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1

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

133 - FML 3 - 18.1W - e.g. 8

Equation of Lines

Find vec form line thru pt ortho to 2 vec form lines

Related Topics

Cross Product

Lines in Rn\reals^nRn

More Cross Product Questions:

Cross Product

Practice Question: Orthogonal Vectors

Practice: Cross Product

Cross Product

Practice Question: Orthogonal Vectors

Practice: Orthogonal Vectors

Practice: Cross Product

Practice Question: Vector Product

Dot and Cross Products: Vector Norm

More Lines in Rn\reals^nRn Questions:

Find vec form line thru pt ortho to 2 vec form lines

133 - FML 3 - 18.1W e.g. 33; 222/262_mid_#2_19.1W_eg5

133 - FML 3 - 18.1W e.g. 50

Lines in $\reals^n$

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

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

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 56

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

$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 58.1

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

133 - FML 3 - 18.1W - e.g. 8

Equation of Lines

Lines in $\reals^n$

More Lines in $\reals^n$ Questions: