10 true or false questions of vector addition and subtraction

Question 1: 1. Vector addition is commutative, meaning $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

True.

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Question 2: 2. The magnitude of the sum of two vectors is always equal to the sum of their magnitudes: $|\vec{A} + \vec{B}| = |\vec{A}| + |\vec{B}|$.

False. This is only true if the two vectors point in the same direction.

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Question 3: 3. Vector subtraction $\vec{A} - \vec{B}$ can be represented as the addition of vector $\vec{A}$ and the negative of vector $\vec{B}$, i.e., $\vec{A} + (-\vec{B})$.

True.

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Question 4: 4. If two non-zero vectors are perpendicular, their sum will always have a magnitude greater than either individual vector's magnitude.

True. By the Pythagorean theorem, $|\vec{A}+\vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2}$, which is greater than $|\vec{A}|$ or $|\vec{B}|$ if both are non-zero.

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Question 5: 5. The resultant vector of adding two vectors can be the zero vector only if the two vectors have the same magnitude and point in opposite directions.

True.

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Question 6: 6. The triangle rule for vector addition states that if two vectors are represented by two sides of a triangle taken in the same order, their sum is represented by the third side taken in the opposite order.

True.

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Question 7: 7. Adding a zero vector to any non-zero vector changes the magnitude and direction of that vector.

False. Adding a zero vector leaves the original vector unchanged in both magnitude and direction.

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Question 8: 8. If $\vec{A} + \vec{B} = \vec{C}$, then it is always true that $\vec{A} = \vec{C} - \vec{B}$.

True. This is a basic property of vector algebra.

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Question 9: 9. The order in which multiple vectors are added affects the final resultant vector.

False. Vector addition is associative, meaning $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.

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Question 10: 10. If two vectors have the same magnitude, their difference will always be zero.

False. Their difference is zero only if they also have the same direction (i.e., they are the same vector). If they have the same magnitude but different directions, their difference will be a non-zero vector.