Vectors

When two vectors of unequal magnitude are added, which of the following statements is true about the magnitude of the resultant vector?

• The resultant vector must be greater than either of the two original vectors.

• The resultant vector must be smaller than both original vectors.

• The resultant vector could be smaller, equal to, or greater than either of the original vectors.

• The resultant vector is always equal to the difference between the two vectors.

Answer explanation

When adding two vectors of unequal magnitude, the resultant can vary based on their direction. It can be smaller, equal to, or greater than either vector, depending on the angle between them.

Which of the following best explains how the direction of the resultant vector is determined when adding two vectors?

• The direction is always along the longer of the two vectors.

• The direction depends only on the angle between the two vectors.

• The direction is along the vector with the larger horizontal component.

• The direction is determined by both the magnitude and direction of the individual vectors.

Answer explanation

The direction of the resultant vector is determined by both the magnitude and direction of the individual vectors. This means that the angle between them and their lengths influence the final direction.

Why is it useful to break a vector into its horizontal and vertical components?

• To simplify the process of vector subtraction.

• To analyze motion or forces in different directions independently.

• To reduce the vector’s magnitude.

• To eliminate the need for vector addition.

Answer explanation

Breaking a vector into horizontal and vertical components allows for independent analysis of motion or forces in different directions, making it easier to understand and solve problems involving vectors.

Which of the following statements is true about the zero vector?

• It has no magnitude but can still have a direction.

• It has no magnitude and no direction.

• It has a magnitude of 1 and can point in any direction.

• It is equal in magnitude to any other vector.

Answer explanation

The zero vector is unique in that it has no magnitude (length) and no direction. This distinguishes it from other vectors, which have both magnitude and direction.

Which of the following is a property of all vectors?

• Vectors can be added together only if they have the same magnitude.

• Vectors are fully described by both magnitude and direction.

• Vectors can only point in two dimensions.

• Vectors cannot be multiplied by scalars.

Answer explanation

The correct choice is that vectors are fully described by both magnitude and direction. This is a fundamental property of vectors, distinguishing them from scalars, which only have magnitude.

If two vectors are added and the resultant vector has a magnitude of zero, what can be inferred about the original vectors?

• The two vectors must have different magnitudes.

• The two vectors must have been in opposite directions with equal magnitudes.

• The two vectors were added at right angles to each other.

• The two vectors must have been identical.

Answer explanation

If the resultant vector has a magnitude of zero, it means the two original vectors must cancel each other out. This occurs when they are equal in magnitude but opposite in direction.

Two vectors are said to be parallel if:

• Their magnitudes are the same.

• Their directions are the same or exactly opposite, regardless of magnitude.

• They have equal horizontal components.

• Their resultant vector is always greater than the individual magnitudes.

Answer explanation

Two vectors are parallel if their directions are the same or exactly opposite, regardless of their magnitudes. This means they can have different lengths but still point in the same or opposite directions.

Two forces act on an object at right angles to each other. Which of the following is true about the resultant force?

• It will always be equal to the sum of the two forces.

• The resultant force will have a direction that lies between the two forces.

• The resultant force will point in the direction of the larger force.

• The two forces will cancel each other out.

Answer explanation

When two forces act at right angles, the resultant force is not simply the sum of their magnitudes. Instead, it has a direction that lies between the two forces, determined by their vector components.

Which of the following is a key difference between scalar and vector quantities?

• Scalars are described by direction, while vectors are described by magnitude.

• Vectors require both magnitude and direction, while scalars only have magnitude.

• Scalars can only be positive, while vectors can be positive or negative.

• Vectors can only be added, while scalars can be added and multiplied.

Answer explanation

The correct choice highlights that vectors require both magnitude and direction, while scalars are defined solely by their magnitude. This distinction is fundamental in physics, as it affects how these quantities are used in calculations.

When two vectors are added, the resultant vector is always larger than either of the original vectors.

• True

• False

Answer explanation

The statement is false because when two vectors are added, the resultant vector can be smaller than either of the original vectors, especially when they are in opposite directions. The magnitude of the resultant depends on the angle between them.