Find the vectors and whose initial and terminal points are given

Question

,
Determine whether the located vectors and are parallel.
(a) , , ,
(b) , , ,
Find and if .
Let , and . Find
b. c. d.
e. f. the angle between and g.
h. i. the unit vector in the direction of
a. Determine such that where .
If and are perpendicular unit vectors, show that .
a. Given that , . Determine the value of for which the vectors and will be mutually perpendicular.
Find a real number such that the vectors and are perpendicular.
Vectors and make an angle of . If and then calculate the angle between the vectors and .
a. Find the projection of vector along vector .
b. Find the cosine of the angle between the vectors and .
c. Find the direction cosines of the vector .
Let , and be vectors in . Then: .
Each of the angles between the vectors is . If , and , find the norm of the vector .
Given three vectors such that if , and . Evaluate .
Prove that if and are vectors in , then
(a) (b)
(c) if and only if and have the same direction.
Given three vectors , and be three non-zero vectors. If , is it necessarily true that ? Justify
Find a unit vector perpendicular to both the vectors:
(a) , (b) and
Find the area of the triangle with vertices , and .
Find the area of the parallelogram two of whose sides are the vectors and .
Find the volume of the parallelepiped whose edges are the vectors , and .
Find the value(s) of so that the plane and are perpendicular.
Let . Then find
parametric equation of the line through the point and parallel to .
the intersection point(s) of and the plane .
Let , be two lines. Then find:
the point of intersection and .
an equation of the plane containing the lines and .
Sketch the graph of the plane.
Find the equation of the plane passing through the points , and .
Find the equation of the line of intersection of the plane and .
Find the equation of the plane passing through the points , and parallel to the line joining the points and .

Answer 1

The tasks you've presented involve various vector operations, including determining vectors from given points, checking for parallelism, calculating angles between vectors, and finding equations for lines and planes. However, to provide a specific solution, I need actual numerical or algebraic inputs for the vectors or points, which are missing in your question.

Please provide the specific vector values or points for the calculations that you would like to perform. Once I have that information, I can assist you with each part of your query.