Adding and subtracting vectors graphically

How are vectors added graphically?

The two vectors and their resultant form three sides of a triangle, so this method is also known as triangle method of vector addition.

NCERT Physics Part I (Class 11), Section 3.4, page 30

In plain words: walking 3 km east and then 4 km north does not put you 7 km from home; it puts you 5 km away, about 53° north of east. To combine vectors you must join the arrows, not add the numbers.

The triangle law

To find A + B, draw A, then draw B starting where A ends. The arrow from the start of A to the end of B is the resultant (NCERT Fig. 3.4, page 29). Moving B there is allowed because it is a free vector: sliding it parallel to itself does not change it. Because the arrows go head to tail, this is also called the head-to-tail method (page 30). (NCERT’s Section 3.4 opens by saying that this law was mentioned in “section 4.2”; it means Section 3.2 (Scalars and vectors, page 28), a leftover from older chapter numbering.)

Vector addition obeys the same two laws as adding numbers (NCERT Eqs. 3.1 and 3.2, page 30):

• commutative: A + B = B + A; adding in either order gives the same resultant;
• associative: (A + B) + C = A + (B + C); for several vectors, chain them head to tail in any order, and the resultant runs from the first tail to the last head.

The length of the resultant is set by the angle between the vectors:

• same direction: |A + B| = A + B, the largest possible;
• opposite directions: |A + B| = |A − B|, the smallest possible;
• at any angle in between, the length lies between these two: |A − B| ≤ |A + B| ≤ A + B (the triangle inequality; NCERT Exercise 3.6, pages 46–47);
• equal magnitudes A at angle θ: the parallelogram is a rhombus, its diagonal bisects θ, so |A + B| = 2A cos(θ/2): 2A at 0°, √2A at 90°, A at 120°, zero at 180°.

For the exact length and direction at any angle, add the components or use the law of cosines given there.

If the chained arrows end where they started, the resultant is zero. This is how forces on a particle in equilibrium behave: drawn head to tail, they close into a triangle or polygon.

The parallelogram law

Draw A and B from a common point O. Through the head of A draw a line parallel to B, and through the head of B a line parallel to A. They meet at S. The diagonal OS is A + B (NCERT Fig. 3.6, page 31). The side of the parallelogram opposite B is a copy of B sitting at the head of A, so this is the triangle law again: the two methods are equivalent (page 30).

The null vector

A + (−A) has zero magnitude. This is the null vector or zero vector, 0 (NCERT Eq. 3.3, page 30). Its properties (NCERT Eq. 3.4):

• A + 0 = A;
• λ0 = 0;
• 0A = 0.

Physically, a null vector is, for example, the displacement of an object that moves from P to P′ and comes back to P (page 30).

Subtracting vectors

Subtraction is defined through addition (NCERT Eq. 3.5, page 30):

Reverse B (that is, multiply it by −1) and add it to A head to tail (NCERT Fig. 3.5, page 30). In the parallelogram of A and B, one diagonal is A + B and the other, from the head of B to the head of A, is A − B (figure above). Check: B + (A − B) = A. For non-zero A and B, |A + B| = |A − B| means the parallelogram is a rectangle and the vectors are perpendicular. If either vector is zero the equality also holds, but an angle with that vector is undefined.

Subtraction is not commutative: B − A = −(A − B), the same length pointing the other way. A common use is the change in a vector, such as Δv = v_final − v_initial (Example 5).

Solved examples

Common mistakes to avoid

Quick revision summary

• Triangle law: B from the head of A; R = A + B from the tail of A to the head of B.
• Parallelogram law: A and B from one point; A + B is the diagonal from that point; the other diagonal is A − B.
• Addition is commutative and associative; |A − B| ≤ |A + B| ≤ A + B; for equal magnitudes, |A + B| = 2A cos(θ/2).
• A + (−A) = 0, the null vector, whose direction cannot be specified.
• A − B = A + (−B); B − A = −(A − B).

Questions students ask

What is the difference between the triangle law and the parallelogram law of vector addition?

The triangle law and the parallelogram law of vector addition give exactly the same resultant; they are two ways of drawing the same sum. The triangle law joins the vectors head to tail, which extends easily to three or more vectors, while the parallelogram law draws both from one point, which matches two forces acting on the same particle and also shows A − B as the second diagonal.

What is the formula for the parallelogram law of vector addition?

For two vectors of magnitudes A and B with an angle θ between them (drawn tail to tail), the parallelogram law gives a resultant of magnitude R = √(A^² + B^² + 2AB cos θ), at an angle α to A with tan α = B sin θ/(A + B cos θ). Use component signs to choose the quadrant of α. If A + B cos θ = 0 and the resultant is non-zero, it is perpendicular to A; a zero resultant has no direction. The triangle law leads to the same formula, since it is the same sum drawn differently. For example, forces of 3 N and 5 N at 60° give R = √(9 + 25 + 15) = √49 = 7 N, at about 38.2° to the 3 N force; the formula is derived in NCERT Example 3.2 (page 34).

What is the polygon law of vector addition?

The polygon law of vector addition states that if several vectors are drawn head to tail, one after another, their resultant is the vector from the tail of the first to the head of the last, the side that closes the polygon. It is the triangle law applied again and again, and the order of the vectors does not change the result, because vector addition is commutative and associative. If the vectors themselves form a closed polygon, their resultant is the null vector.

Can the resultant of two vectors be smaller than each of them?

Yes, the resultant of two vectors can be smaller than each of them, but only when they largely oppose each other: with A the smaller magnitude, the condition is cos θ < −B/(2A), so the angle between them must be more than 120° and B less than 2A. For example, forces of 5 N and 7 N at 150° give R = √(25 + 49 + 70 cos 150°) ≈ 3.7 N, less than both. At 90° or less, the resultant is always longer than the larger of the two.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Two vectors of magnitudes 6 and 8 units are added. What are the largest and smallest possible magnitudes of the resultant?

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   Largest 6 + 8 = 14 units (same direction); smallest 8 − 6 = 2 units (opposite directions).

2. Q2. A man walks 30 m north, then 40 m east. How far is he from his starting point?

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   √(30^² + 40^²) = 50 m, towards the north-east (about 53° east of north).

3. Q3. Is A − B the same as B − A?

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   In general, no: B − A = −(A − B). If A ≠ B, the differences have opposite directions and equal magnitudes. If A = B, both differences are the zero vector and are equal.