Question

Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-5, 4, -1), Q = (-3, 6, 1), and R = (-3, 6, 2).

Answer #1

Find two unit vectors orthogonal to ?=〈−4,4,5〉a=〈−4,4,5〉 and
?=〈4,0,−4〉b=〈4,0,−4〉
Enter your answer so that the first vector has a positive first
coordinate

Find an equation of a sphere with the given radius r
and center C. (Use (x,
y, z) for the coordinates.)
r =
7; C(3, −5, 2)
Find the angle between u and
v, rounded to the nearest tenth degree.
u = j + k,
v = i +
2j − 5k
Find the angle between u and
v, rounded to the nearest tenth degree.
u = i + 4j −
8k, v =
3i − 4k
Find a vector that...

Let P be the plane given by the equation 2x + y − 3z = 2. The
point Q(1, 2, 3) is not on the plane P, the point R is on the plane
P, and the line L1 through Q and R is orthogonal to the plane P.
Let W be another point (1, 1, 3). Find parametric equations of the
line L2 that passes through points W and R.

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

Find a unit vector orthogonal to <1,3,-1> and
<2,0,3>

1.
(1 point)
Find the distance the point P(1, -6, 7), is to the plane through
the three points
Q(-1, -1, 5), R(-5, 2, 6), and S(3, -4, 8).
2.
(1 point) For the curve given by
r(t)=〈−7t,−4t,1+7t2〉r(t)=〈−7t,−4t,1+7t2〉,
Find the derivative
r′(t)=〈r′(t)=〈 , , 〉〉
Find the second derivative
r″(t)=〈r″(t)=〈 , , 〉〉
Find the curvature at t=1t=1
κ(1)=κ(1)=

Find a unit vector orthogonal to the vectors ? = 〈1,5, −2〉 and
?⃗ = 〈−1,3,0〉.

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

Show complete solution.
1. Find two unit vectors that are parallel to the ?? −plane and
are orthogonal to the vector ? = 3? − ? + 3?.

3. Consider the plane with a normal vector 〈2, 5, −1〉 which
contains the point (3, 5, −1), and the plane containing the points
(0, 2, 1), (−1, −1, 1), and (1, 2, −2). Determine whether the
planes are parallel, orthogonal, or neither.

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