Understanding Vector and Scalar Projections

What do two non-parallel vectors in 3D space define?

• A cube

• A point

• A plane

• A line

What is the vector projection of vector v onto vector w?

• The difference between v and w

• The component of v in the direction of w

• The sum of v and w

• The perpendicular component of v to w

How is the scalar projection of a vector defined?

• As the angle between the vectors

• As the cross product of the vectors

• As the magnitude of the vector projection

• As the dot product of the vectors

What does a negative scalar projection indicate?

• The vectors are perpendicular

• The projection vector is in the opposite direction

• The vectors are parallel

• The projection vector is zero

Which vector is in the denominator of the projection formula?

• The difference of both vectors

• The sum of both vectors

• The vector onto which another is projected

• The vector being projected

What is the vector projection of v onto w if v = (5, -2, 1) and w = (3, 1, -4)?

• (27/26, -18/13, 9/26)

• (18/13, 9/26, 27/26)

• (9/26, 27/26, -18/13)

• (27/26, 9/26, -18/13)

How do you calculate the magnitude of a vector?

• By multiplying all components

• By taking the square root of the sum of the squares of its components

• By subtracting the smallest component from the largest

• By adding all components

What is the scalar projection of w onto v if w = (-3, -1, 0) and v = (1, 4, 2)?

• √21/3

• -√21/3

• 7/√21

• -7/√21

What is the purpose of rationalizing a fraction in scalar projection?

• To simplify the numerator

• To eliminate the square root from the denominator

• To make the fraction larger

• To change the sign of the fraction

What is the result of the dot product of two perpendicular vectors?

• A negative number

• A positive number

• Zero

• The product of their magnitudes