Subtraction of Vectors: How to Subtract Two Vectors
Physics · Motion In A Plane · NEET
To subtract vector B from vector A, you do NOT subtract like numbers. Instead you reverse B (flip its direction to get -B) and add it to A: A - B = A + (-B). Memory hook: "To subtract, flip and add" — turn B around, then use the same triangle/head-to-tail rule you already know for addition.
To find A - B, reverse B to get -B (dashed red), place its tail at the head of A, and draw the green resultant from the tail of A to the head of -B. The magnitude uses a minus sign: |A - B| = sqrt(A^2 + B^2 - 2AB cos(theta)).
Your doubts, answered
Is A - B the same as B - A for vectors?
No. In vector subtraction A - B and B - A have the SAME magnitude but OPPOSITE directions. So A - B = -(B - A). This is different from vector addition, where A + B = B + A (order does not matter). Subtraction is NOT commutative — always keep track of which vector is first.
How do I actually subtract two vectors that point in different directions?
Three steps. (1) Reverse the vector you are subtracting: -B has the same length as B but points the opposite way. (2) Place the tail of -B at the head of A (head-to-tail). (3) Draw the resultant from the tail of A to the head of -B. That resultant is A - B. This is just the triangle law of addition applied to A and -B.
Why does 'change in velocity' or 'change in momentum' need vector subtraction?
Change means final minus initial: delta-v = v_final - v_initial. Because velocity and momentum are vectors, you must subtract them as vectors, not as plain numbers. A body moving at the same speed but in a new direction still has a change in velocity, so its acceleration and the force on it are not zero. This is exactly why a car turning a corner needs a force even at constant speed.
What is the formula for the magnitude of A - B?
|A - B| = sqrt(A^2 + B^2 - 2AB cos(theta)), where theta is the angle between A and B. Notice the MINUS sign in front of 2AB cos(theta) — for addition |A + B| uses a PLUS sign. When theta = 90 degrees, cos(theta) = 0, so |A - B| = |A + B| = sqrt(A^2 + B^2).
Can the subtraction of two vectors give a zero vector?
Yes. If A and B are equal vectors (same magnitude AND same direction), then A - B = 0, the null vector. This is the only case where the difference of two non-zero vectors is zero. If they have equal magnitude but different directions, A - B is non-zero.
⚠️ The NEET trap ✗ For a body that keeps the same SPEED but changes direction (like turning a corner), students say the change in velocity is zero because the speed did not change. ✓ Velocity is a vector, so delta-v = v_final - v_initial must be done by vector subtraction. Same speed in a new direction gives a NON-zero change in velocity, so there IS acceleration and force. In the NEET 2023 football-player question, turning from south to east gives a force directed north-east, not zero. 🧠 Same speed does NOT mean same velocity. Direction change alone = change in velocity.
Real NEET questions
2023
A football player is moving southward and suddenly turns eastward with the same speed to avoid an opponent. The force that acts on the player while turning is directed:
A · Along eastward
B · Along northward
C · Along north-east ✓
D · Along south-west
Solution: Force is along the change in momentum: delta-p = p_final - p_initial (a vector subtraction). Take east as +x and north as +y. Initial motion is southward, so p_initial = -p (north-hat), i.e. it points south. Final motion is eastward, so p_final = p (east-hat). Then delta-p = p(east) - (-p(north)) = p(east) + p(north) = p(east-hat + north-hat). Adding an eastward and an equal northward vector gives a resultant at 45 degrees between them = north-east. So the force points north-east.
2016
If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, the angle between these vectors is:
A · 0 degrees
B · 90 degrees ✓
C · 45 degrees
D · 180 degrees
Solution: Set |A + B| = |A - B|. Square both sides: A^2 + B^2 + 2AB cos(theta) = A^2 + B^2 - 2AB cos(theta). Cancel A^2 + B^2 from both sides: 2AB cos(theta) = -2AB cos(theta), so 4AB cos(theta) = 0. Since A and B are non-zero, cos(theta) = 0, giving theta = 90 degrees. The sum and difference are equal in magnitude only when the two vectors are perpendicular.
Solved Motion In A Plane NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
A - B is defined as A + (-B): reverse vector B to get -B, then add it to A using the triangle or parallelogram rule. It is never a plain arithmetic subtraction of magnitudes unless the vectors are parallel.
Is vector subtraction commutative?
No. A - B is not equal to B - A. They have equal magnitude but opposite direction, so A - B = -(B - A). Only vector addition is commutative.
What is the magnitude of the difference of two vectors?
|A - B| = sqrt(A^2 + B^2 - 2AB cos(theta)), where theta is the angle between A and B. The minus sign in front of the 2AB cos(theta) term is what distinguishes it from the addition formula.
When is the difference of two vectors a null vector?
When the two vectors are equal — same magnitude and same direction. Then A - B = 0, the zero vector. This is used to test equality of vectors.
Why is vector subtraction important for NEET?
Almost every 'change in' quantity uses it: change in velocity, change in momentum, and relative velocity all come from subtracting vectors. NEET regularly asks about the direction of the force when a body turns, which is pure vector subtraction.
