Question

Find an equation of a sphere with the given radius *r*
and center *C*. (Use (* x*,

Find the angle between **u** and
**v**, rounded to the nearest tenth degree.
**u** = **j** + **k**,
**v** = **i** +
2**j** − 5**k**

Find the angle between **u** and
**v**, rounded to the nearest tenth degree.
**u** = **i** + 4**j** −
8**k**, **v** =
3**i** − 4**k**

Find a vector that is perpendicular to the plane passing through the three given points. P(3, 4, 5), Q(4, 5, 6), R(4, 7, 6)

Find a vector that is perpendicular to the plane passing through the three given points. P(1, 1, 8), Q(2, 2, 0), R(0, 0, 0)

Find a vector that is perpendicular to the plane passing through the three given points. P(3, 0, 0), Q(0, 2, −5), R(−2, 0, 6)

Answer #1

2. Given ⃗v = 〈3,4〉 and w⃗ = 〈−3,−5〉 find
(a) comp⃗vw⃗
(b) proj⃗vw⃗
(c) The angle 0 ≤ θ ≤ π (in radians) between ⃗v and w⃗.
1. Let d--> =2i−4j+k. Write⃗a=3i+2j−6k as the sum
of two vectors⃗v,w⃗, where⃗v is perpendicular
to d--> and w⃗ is parallel to
d-->.

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 an equation for each of the following planes. Use x, y and
z as the variables.
a) An equation of the plane passing through the points (1,−1,1),
(0,−2,−1) and (−4,0,6)
b) An equation of the plane consisting of all points that are
equidistant (equally far) from (−3,−5,−1) and (4,−1,−3)
c) An equation of the plane containing the line
x(t)= [0, -1, 1] + t[0, 4, -1] and is
perpendicular to the plane 3y − 4z = −7

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).

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...

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.

Consider the graph of the function z=2x-3y+c in a plane.
In case of c=4, find three distinct points P, Q, R such that the
vector Q-P is not a scalar multiple of R-P.

Find an equation of the tangent plane to the given parametric
surface at the specified point. x = u + v, y = 6u^2, z = u − v; (2,
6, 0)

Find an equation of the sphere with center
(2, −6, 4)
and radius 5.
Use an equation to describe its intersection with each of the
coordinate planes. (If the sphere does not intersect with the
plane, enter DNE.)
intersection with
xy-plane
intersection with
xz-plane
intersection with
yz-plane

Find an equation of the line that satisfies the given
conditions. Through (5, 7); perpendicular to the line y = 4
Find an equation of the line that satisfies the given
conditions. Through (−5, −7); perpendicular to the line passing
through (−2, 5) and (2, 3)

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