Question

Two 3D vectors, A and B , are represented in terms of the Cartesian basis unit vectors, i , j , k as follows:

A=2i-3j+k

B=10i+9j-4k

1) plot A and B in a Cartesian coordinate system

2) calculate A + B

3) calculate A · B using the scalar product formula

4) calculate the magnitudes of A and B

5) calculate the angle between A and B

6) calculate A x B using the determinant formula

7) show that the new vector, w = A x B , is perpendicular to both A and B

Answer #1

A) Find the angle between the vectors 8i-j + 4k and 4j + 2k
B) Find c so that the vectors 2i-3j + ck and 2i-j + 4k are perpendicular
C) Find the scalar projection and the vector projection of 2i +
4j-4 on 3i-3j + k

Two vectors given below, A and B, are located in a standard 3-D
cartesian coordinate system: A = 5i + 2j - 4k B = 2i + 5j + 5k a.
Find the magnitude of the sum of A and B. b. Find the dot product
of A and B. What does this result tell you about A and B? c. Find a
vector C, with non-zero magnitude, that is perpendicular to both A
and B.

1) Find the angle θ between the vectors a=9i−j−4k and
b=2i+j−4k.
2) Find two vectors v1 and v2 whose sum is <-5,
2> where v1 is parallel to <-3 ,0> while v2 is
perpendicular to < -3,0>

Using MATLAB, create three vectors a = 4i, b = 2i - 4j, and c =
-2i + 3k, where i, j and k are unit vectors of three axes in
Cartesian coordinate system. Compute |?∙(?×?)| using the predefined
MATLAB commands and show that it is the volume of a parallelepiped
defined by three vectors a, b and c.

The three following coordinate vectors are given in unitary
coordinates (in [m]):
a = (5; 0; 0)
b = (-1; 4; 1)
c = (0; 1; 3)
a) Determine the new coordinate system, giving |a|, |b|, |c|,
alpha, beta and gamma. Use vector operations to obtain the
values.
b) Determine the metric matrix for this coordinate system, and
the volume of the parallelepiped spanned by a, b, c. For the volume
calculation, use the determinant of the metric matrix.
c)...

Given the vectors in unit vector notation:
A=(-3.0m)i+(4.0m)j
B=(4.0m)i+(-7.0m)j
Calculate the magnitude and the directional angle of the resultant
of r=a+b with respect to the + x axis.

Your task will be to derive the equations describing the
velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general
2D motion starting from a general
position vector. Then use these expressions to simplify to the case
of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate
system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t)...

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