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Mathematics-Online test:

# Vector Calculus, Test 3

This test contains problems (P) with different versions (V).

 Shown: P1 V3 P2 V- P3 V1 P4 V- P5 V- P6 V- P7 V- Select: randomly 1 2 3 4 5 6 7 - randomly 1 2 - - - -

test with versions selected.

Problem 1:
Solve the following linear system of equations:

Solution:

,      ,

(Specify the solution rounded on three decimal places)

Problem 2:
Consider a parallelogram . Denote by the centre of line segment and let denote the point of intersection of the line segment and the diagonal .

a)
At what ratio does partition the diagonal ?
b)
What is the ratio of the area of the parallelogram to the area of the triangle ?

Solution:

a)
The ratio is : .
b)
The ratio area : area is : .

Problem 3:
Given vectors

Find
a) ,         b) ,         c) ,         d) .

Solution:
a),         b) ,, ,        c) ,, ,        d)

Problem 4:

Consider the triangle with vertices

Compute the length of all sides, the angle and the aera of the triangle.

Antwort:
,     ,     ,     .
Square of the area of the triangle: .

Problem 5:
Given the points , and in . Let be the line through and , and let be the line through with direction

Find the distance of from , and the distance between and .

Solution:

Distance of from : .

Distance between and : .

Problem 6:
Given the following plane in

a)
Let be the plane through point , parallel to . Find the equation describing the plane .

b)
Which point on plane has minimal distance from point . What is the minimal distance?

c)
Show that point lies in the plane , and that the points form an equilateral triangle. Find the lenghts of the sides, all interior angles, and the area of the triangle.

Solution:

a)
Complete the missing coefficients of the equation of : .
b)
Point , , , distance: .
c)
squared sides: , , .
, , .
Area of triangle: /     (given as completely reduced fraction).

Problem 7:
Let be the line given by

and let be the point .
a)
Calculate the distance between and .
b)
Determine the defining equation of the plane spanned by the points , and .
c)
Let be the plane spanned by and . At which angle do and intersect each other?
d)
Calculate the volume of the parallelepiped spanned by , , and .

Solution:

a)
Squared distance: .
b)
: .
c)
Angel of intersection: /.
d)
Volume: .

 () automatically generated 8/11/2017