Vectors

Question 1

What is a vector?

Answer 1

A vector is a quantity that has both magnitude and direction.

Question 2

True or False: Vectors can be added geometrically using the head-to-tail method.

Answer 2

True

Question 3

Fill in the blank: The magnitude of a vector ( mathbf{a} = (x, y) ) is given by ( ||mathbf{a}|| = sqrt{____} ).

Answer 3

x^2 + y^2

Question 4

What is the result of adding two vectors ( mathbf{u} = (2, 3) ) and ( mathbf{v} = (5, 7) )?

Answer 4

(7, 10)

Question 5

What is a unit vector?

Answer 5

A unit vector is a vector with a magnitude of 1.

Question 6

True or False: The dot product of two perpendicular vectors is zero.

Answer 6

True

Question 7

What is the formula for the dot product of vectors ( mathbf{a} = (a_1, a_2) ) and ( mathbf{b} = (b_1, b_2) )?

Answer 7

a_1 * b_1 + a_2 * b_2

Question 8

If vector ( mathbf{a} = (3, 4) ), what is the unit vector in the direction of ( mathbf{a} )?

Answer 8

(0.6, 0.8)

Question 9

Fill in the blank: The cross product of two vectors in 3D results in a vector that is ____ to the plane formed by the two original vectors.

Answer 9

perpendicular

Question 10

What is the formula for the cross product of vectors ( mathbf{a} = (a_1, a_2, a_3) ) and ( mathbf{b} = (b_1, b_2, b_3) )?

Answer 10

(a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)

Question 11

True or False: The magnitude of the cross product of two vectors can be calculated as ( ||mathbf{a}|| ||mathbf{b}|| sin( heta) ).

Answer 11

True

Question 12

What is the angle between two vectors if their dot product is equal to zero?

Answer 12

90 degrees

Question 13

If ( mathbf{a} = (1, 2) ) and ( mathbf{b} = (3, 4) ), what is the angle between them in degrees?

Answer 13

approximately 16.26 degrees

Question 14

What is a position vector?

Answer 14

A position vector represents the position of a point in space relative to an origin.

Question 15

Fill in the blank: The vector ( mathbf{a} = (x_1, y_1) ) is parallel to the vector ( mathbf{b} = (x_2, y_2) ) if there exists a scalar ( k ) such that ( mathbf{a} = k cdot mathbf{b} ).

Answer 15

scalar

Question 16

What does it mean for two vectors to be orthogonal?

Answer 16

Two vectors are orthogonal if their dot product is zero.

Question 17

If vector ( mathbf{a} = (4, 3) ), what is the direction angle ( heta ) with respect to the positive x-axis?

Answer 17

approximately 36.87 degrees

Question 18

True or False: The resultant vector of two vectors can be found by subtracting their magnitudes.

Answer 18

False

Question 19

What is the geometric interpretation of the scalar triple product?

Answer 19

The scalar triple product represents the volume of the parallelepiped formed by three vectors.

Question 20

Fill in the blank: The angle between two vectors can be found using the formula ( cos( heta) = rac{____}{||mathbf{a}|| ||mathbf{b}||} ).

Answer 20

(mathbf{a} cdot mathbf{b})

Question 21

What is the result of the cross product of two parallel vectors?

Answer 21

The zero vector.

Question 22

What is the component form of vector ( mathbf{a} ) if ( mathbf{a} ) has a magnitude of 5 and makes an angle of 30 degrees with the positive x-axis?

Answer 22

(5 * cos(30), 5 * sin(30))

Question 23

True or False: Vectors can be multiplied by scalars, which affects their magnitude but not their direction.

Answer 23

False

Question 24

What is the equation of a line in vector form?

Answer 24

The equation is given by ( mathbf{r} = mathbf{a} + tmathbf{b} ), where ( mathbf{a} ) is a point on the line and ( mathbf{b} ) is the direction vector.

Question 25

Fill in the blank: A vector can be represented in component form as ( mathbf{a} = (____, ____) ).

Answer 25

(a_1, a_2)