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

Related Topics

Wize University Linear Algebra Textbook > Products of Vectors

Projection (Proj) and Perpendicular (Perp)

6 Activities

Wize University Linear Algebra Textbook > Equations of Lines and Planes

Planes in R3\reals^3

6 Activities

Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)} and P2=(1,−2,3)\bcb{P_2 = (1,-2,3)} and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}, find the point Po\bcb{P_o} on the plane Π: 2x−3y+z=1\bcb{\Pi: \ 2x - 3y + z = 1} that is closest to the point P2\bcb{P_2}.
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\rangleonto a⃗=⟨−1,4,2⟩\vec{a}=\langle -1,4,2\rangle.
Find the vector projection of [3,2][3,2] onto the vector [3,6]\left[3,6\right]
133 - FML 3 - 18.1W - e.g. 9
Find the vector-component ı^∣∣d⃗ \ihat_{||_{\vd}} \, of ı^\bcb{\boldsymbol{ \ihat}} in the direction of the line given by y=mx\bcb{\boldsymbol{ y=mx}}. Repeat for ȷ^∣∣d⃗ \jhat_{||{\vd}} \, , 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>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}
Find the point on the line L\bcb{L} closest to the point P2\bcb{P_2}.
133 - FML 3 - 18.1W e.g. 58.2
Given the vector v⃗=<1,1,2>\bcb{\vec{v} = \left< 1, 1, 2 \right>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}
Find the minimum distance of P2\bcb{P_2} to the line L\bcb{L} found above.
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>} onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
133 - FML 3 - 18.1W e.g. 43
Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>} and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}, find the vector projection of u⃗\bcb{\vec{u}} onto v⃗\bcb{\vec{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)}, Q=(−5,−6,−3)\bcb{Q = (-5,-6,-3)}, R=(−6,−4,−6)\bcb{R = (-6,-4,-6)} and S=(−2,−3,−4)\bcb{S = (-2, -3, -4)}. Let l1\bcb{l_1} be the line passing through P\bcb{P} and Q\bcb{Q}, and let l2\bcb{l_2} be the line passing through R\bcb{R} and S\bcb{S}. Find the distance between R\bcb{R} and l1\bcb{l_1}.
133 - FML 3 - 18.1W e.g. 36
For two lines having directional vectors v1\mathbf{v}_1 and v2\mathbf{v}_2, 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), 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|
Find the equations of the line passing through the point (2,3,−4)\bcb{(2, 3, -4)} parallel to the vector i−4k\bcb{\mathbf{i} - 4\mathbf{k}} ; 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. 48
Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>} onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 43
Given u⃗=<8,2,2>\bcb{\vec{u} = \left< 8, 2, 2 \right>} and v⃗=<2,−1,−1>\bcb{\vec{v} = \left< 2, -1, -1 \right>}, find the vector projection of u⃗\bcb{\vec{u}} onto v⃗\bcb{\vec{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>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}, find the minimum distance of P2\bcb{P_2} to the line L\bcb{L} that is parallel to v⃗\bcb{\vec{v}} that passes through P1\bcb{P_1}.
$\tkct{cut from 19.4F}$ 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)} and P2=(1,−2,3)\bcb{P_2 = (1,-2,3)} and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}
Find the minimum distance between the point P2\bcb{P_2} and the plane Π: 2x−3y+z=1\bcb{\Pi:\ 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>} and the points P1=(0,−1,0)\bcb{P_1 = (0,-1,0)} and P2=(−1,1,0)\bcb{P_2 = (-1,1,0)}, find the point on
the line L\bcb{L} (which is parallel to v⃗\bcb{\vec{v}} that passes through P1\bcb{P_1}) that is closest to the point P2\bcb{P_2}.
More Planes in R3\reals^3 Questions:
Write down the equation of a plane defined by this parametric equation: P: (-2,0,5)+s(3,1,2)+t(-2,2,0)
Consider the two lines, L_1:(1,0,0)+s(1,1,1) and L_2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Write down equation of Line L: (1,3,5)+t(-4,1,3) as the intersection of two planes.
Find a point whose distance to the plane 3x+4y−10z=303x+4y-10z=30 is equal to 10.
Consider two planes with equations 3x+2y−z=8 and ax−6y+bz=8.3x+2y-z=8\ and\ ax-6y+bz=8.
a) For what values of a and b are two planes parallel?
b) For what values of a and b is the angle between two planes 60 degrees?
If two planes 2x+ay−2z=2y+62x+ay-2z=2y+6 and x=y+z+3x=y+z+3 are parallel, at which point does the first plane intersect with x-axis.
Consider two lines: L1:(1,0,0)+s(1,1,1) and L2:(-1,2,3)+t(4,0,c).
a) For what value of c two lines intersects.
b) Find the equation of the plane containing both lines.
Find gen form plane thru pt parallel to 2 vecs
Given that the plane PP passes through the point (1,3,−1)\left(1,3,-1\right) and is parallel to both vectors
𝑢⃗=(0,1,1)𝑢⃗ = (0,1,1) and 𝑣⃗=(3,2,1),\vec𝑣 = (3,2,1),find the standard form equation of the plane.
(type your answer in the form ax+by+cz=d, with no spaces)
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 59.1
Given the points P1=(0,0,1)\bcb{P_1 = (0,0,1)} and P2=(1,−2,3)\bcb{P_2 = (1,-2,3)} and the vector v⃗=<2,−3,1>\bcb{\vec{v} = \left< 2, -3, 1 \right>}
Find the equation of the plane Π\bcb{\Pi} perpendicular to v⃗\bcb{\vec{v}}, containing the point P1\bcb{P_1}.
Planes in $\reals^3$
Practice: Normal Vector
Which of the following vectors is normal to the plane 3x+z=73x+z=7?
A plane is parallel to the vector v=⟨1,1,0⟩v = \langle1, 1, 0\rangle. If it passes through points (1,2,−1)(1,2,-1) and (2,1,0)(2,1,0) then find the equation of the plane.
Which one of the following is a vector that is orthogonal to the line that passes through the points P(1,−1,0)P\left(1,-1,0\right) and Q(5,0,−3)Q\left(5,0,-3\right)?
Which of the following is a vector orthogonal/ perpendicular to the plane 2x−z=32x-z=3?
133 - FML 3 - 18.1W e.g. 30
Find the equation of a plane passing through the point (2,0,1)\bcb{(2, 0, 1)}, perpendicular to the straight line passing through the points (1,1,0)\bcb{(1, 1, 0)} and (4,−1,−2)\bcb{(4, -1, -2)}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.7
Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2} and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}, and the lines
L1 : {x=3t+2y=−2t+1z=t, L2 : {x=−2t+1y=t+1z=t+3, L3 : {x=t+3y=−tz=2t+1\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.},
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 60.4
Given the planes
π1 : 4x−3y+z=2\bcb{\pi_1 \, : \, 4x - 3y + z = 2} and π2 : x+2y+2z=0\bcb{\pi_2 \, : \, x + 2y + 2z = 0}, and the lines
L1 : {x=3t+2y=−2t+1z=t, L2 : {x=−2t+1y=t+1z=t+3, L3 : {x=t+3y=−tz=2t+1\bcb{L_1 \, : \, \left\{ \begin{matrix} x & = & 3t + 2 \\ y & = & -2t + 1 \\ z & = & t \end{matrix} \right., ~ L_2 \, : \, \left\{ \begin{matrix} x & = & -2t + 1 \\ y & = & t + 1 \\ z & = & t + 3 \end{matrix} \right., ~ L_3 \, : \, \left\{ \begin{matrix} x & = & t + 3 \\ y & = & -t \\ z & = & 2t +1 \end{matrix} \right.},
The following linear equations represent planes in R3R^3. Describe the intersection of these planes (if any).
2x+y−z=3x−y−z=33x+z=6\begin{array}{c} 2x+y-z=3\\ x-y-z=3\\ 3x+z=6 \end{array}
Planes in $\reals^3$
Practice: Planes in R3\colorOne{\reals^3} (Tricky)
Consider the points A(−1,3,2)A(-1,3,2), B(−3,5,0)B(-3,5,0), and C(xC,yC,zC)C(x_C,y_C,z_C), and suppose we know the vector AC→=⟨3,−4,−5⟩\overrightarrow{AC}=\lang 3,-4,-5\rang.
Find the equation of the plane containing point CC, and perpendicular to the line passing through points AA and BB.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane 2x+y+3z=6\bcb{2x + y + 3z = 6}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 57
Find the equation of a plane that contains the three points P1=(1,−2,3)\bcb{P_1 = (1,-2, 3)}, P2=(0,−1,4)\bcb{P_2 = (0, -1, 4)} and P3=(4,3,−2)\bcb{P_3 = (4,3,-2)}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 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>} that contains the point P=(−1,1,2)\bcb{P = (-1, 1, 2)}.
133 - FML 3 - 18.1W e.g. 31
Find a point on, and a vector normal to, the plane 2x+y+3z=6\bcb{2x + y + 3z = 6}.
Planes in R^3
Which one of the following is the standard form equation of the plane in R3\reals^3 which contains the lines (x,y,z)=(2,0,−1)+t(2,−5,−1)(x,y,z)=(2,0,-1)+t(2,-5,-1) and (x,y,z)=(3,1,2)+t(3,−5,1)(x,y,z)=(3,1,2)+t(3,-5,1)