Vector Operations Mastery

What is the result of adding the vectors (3, 4) and (1, 2)?

• (2, 3)

• (4, 6)

• (5, 6)

• (3, 5)

If vector A = (2, 3) and vector B = (4, -1), what is A + B?

• (2, 4)

• (6, 2)

• (6, -1)

• (5, 3)

Explain the commutative property of vector addition.

• A + B = A * B for any vectors A and B

• The commutative property of vector addition states that A + B = B + A for any vectors A and B.

• A + B = A - B

• A + B = 0 for any vectors A and B

What is the scalar multiplication of vector (2, 5) by 3?

• (6, 10)

• (2, 15)

• (3, 5)

• (6, 15)

How does scalar multiplication affect the direction of a vector?

• Scalar multiplication always increases the magnitude of a vector.

• Scalar multiplication can keep the direction the same (positive scalar), reverse it (negative scalar), or eliminate it (zero scalar).

• Scalar multiplication has no effect on the vector's magnitude or direction.

• Scalar multiplication can only change the magnitude, not the direction.

Calculate the dot product of vectors (1, 2, 3) and (4, 5, 6).

• 36

• 30

• 28

• 32

What is the geometric interpretation of the dot product?

• The dot product indicates the distance between two points in space.

• The dot product represents the magnitude of one vector in the direction of another, scaled by the cosine of the angle between them.

• The dot product is the sum of the magnitudes of two vectors.

• The dot product measures the area of the parallelogram formed by two vectors.

If vector A = (1, 0, 0) and vector B = (0, 1, 0), what is A × B?

• (0, 0, 1)

• (1, 1, 0)

• (0, -1, 0)

• (0, 0, 0)

Explain the properties of the cross product.

• The cross product is commutative and results in a vector with the same direction as the inputs.

• The cross product of two vectors results in a vector that is perpendicular to both, with properties of non-commutativity, distributivity, scalar multiplication, and a magnitude related to the sine of the angle between them.

• The cross product results in a scalar value that is the sum of the two vectors.

• The cross product can only be performed on three-dimensional vectors.

What is the magnitude of the vector (3, 4)?

• 6

• 5

• 4

• 7