Calculate the volume of the parallelepiped spanned

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Reference no: EM132363420

Calculus Assignment

1. Consider the vectors a^→ = 2i^→ + 4j^→- 6k^→, b^→ = -3i^→ + k^→, and c^→ = 3j^→ - 2k^→.

[1] a) Find the magnitude of the vector a^→.

[1] b) Find the vector s = a^→ + b^→ written as an algebraic vector in components form.

[1] c) Find the magnitude of the vector d^→= c^→ - b^→ .

2. Consider the parallelogram ABCD where A(0,1,2), B(1,2,3) , and C(0,3,5) .

[2] a) Find the coordinates of the vertex D .

[2] b) Find the area of the triangle Δ ABC

[2] b) Find the angle ∠B .

3 Consider the following vectors: a^→ = i^→ - 2j^→+ 3k^→, b^→ = 2i^→ - 3k^→, and c^→ = -3i^→ + 2j^→

Compute the required operations.

[1]a) a^→ - 2b^→+ 3c

[1] b) a^→ . b^→

[1] c) b^→ x c^→

[2] d) (a^→ x b^→) . c^→

4. Consider the vector a^→ = (-2, 3, -1) .

[1]a) Find the magnitude of the vector a^→.

[1] b) Find an unit vector parallel to the vector a^→.

[3] c) Find the direction angles (the angles between the vector a^→ and the coordinate axes).

5. Find an unit vector perpendicular to both vectors a(2, -3, 0) and b(0, 1, -2)

6. Find the work done by the force F=(-2,3,-5)when the object is moved from the point A(0,1,4)to the point B(-3,0,5).

7. A wrench 30cm long is used to loose a bolt by applying a force of 20N (see the figure below). Find the magnitude of the torque.

8. Is Dot product a vector or a scalar product? Explain.

9. If two vectors are pointing in opposite directions, what is the value of a^→ . b^→? Explain.

10. Explain what it means if a^→ . b^→ = 0

11.A block is dragged across a floor with a force of 200N that acts parallel to the floor, (in direction of displacement), over a distance
of 3m. Calculate the work done.

12.A block is dragged across a floor with a force of 200N that acts at an angle of 30 degrees to the direction of displacement), over a distance of 3m. Calculate the work done. Explain the difference between this calculation and that of the previous question in qualitative terms.

13.A block is dragged across a floor with a force of 200N that acts at an angle of 90 degrees to the direction of displacement), over a distance of 3m. Calculate the work done. Explain the difference between this calculation and that of the previous question in qualitative terms.

14.What is the result of a^→ . a^→? Explain its meaning.

15.Identify the type of answer that results from each operation. If the operation is not possible, indicate why.,

a. k^→x (a^→ x b^→)

b. (a^→ x b^→)

c. k^→. (a^→ x b^→)

16.The cross product is anti-commutative. This means that

a^→x b^→ = -b^→ x a^→. Show this is true for a^→=(2, 3, 4) , b^→= (-5,-4, -3)

17.The volume of a parallelepiped is given by the following formula:

a. c^→. (a^→ x b^→)

Calculate the volume of the parallelepiped spanned by the vectors:

a^→= (1,2,3) ,b^→= (9,10,11), c^→= (11,6,10)

18.Calculate the volume of the of the parallelepiped spanned by the following vectors. Interpret your result.

a^→= (1, 2 ,3) , b^→= (9,10,11), c^→= (11,14,17)

Reference no: EM132363420

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