A vector with zero magnitude and indeterminate direction is called a ___ vector.

Null vector

True or False: The addition of vectors is associative, meaning (a + b) + c = a + (b + c).

True

The resultant R of two vectors A and B, represented by the parallelogram law of addition, can be expressed as R = ___? (in terms of |A|, |B|, and θ)

R = √(|A|² + |B|² + 2|A||B|cosθ)

If two vectors are equal, then they have the same magnitude and ___ direction.

Same

Fill in the blank: The scalar (dot) product of two unit vectors a and b is equal to ___, where θ is the angle between them.

Cosθ

True or False: The cross product of two non-parallel vectors is always zero.

False; the cross product is non-zero and produces a vector perpendicular to both a and b.

The vector resolution of a vector R in the direction of vectors A and B can be expressed as R = ___? (using components).

R = xA + yB, where x and y are scalars.

The angle between two vectors a and b can be calculated using the formula ___?

Cosθ = (a · b) / (|a||b|)

If vectors A and B are collinear, then their cross product A × B is equal to ___?

Zero

Fill in the blank: The position vector of a point P with respect to the origin O is denoted as ___?

OP

True or False: The scalar triple product [A B C] is zero if any two vectors A, B, or C are parallel.

True

The vector moment (torque) M of a force F acting at point A about point O is given by M = ___?

R × F, where R is the position vector OA.

For two vectors A and B, if θ = 90°, then the scalar product A · B equals ___?

Zero

Fill in the blank: The distance of a point from a line is given by the formula ___?

D = |(R - A) · B| / |B|, where R is the position vector of the point, A is a point on the line, and B is the direction vector of the line.

If the vectors A, B, and C are linearly independent, then the determinant formed by their components is ___?

Non-zero