Multiplying a vector by a real number

What happens when a vector is multiplied by a number?

Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A

NCERT Physics Part I (Class 11), Section 3.3, page 29

In plain words: multiplying by 3 makes the arrow three times as long; multiplying by ½ halves it; a minus sign turns it round. The line the arrow lies along never changes.

Positive and negative factors

NCERT’s examples (page 29, Fig. 3.3):

• 2A points the same way as A and is twice as long;
• −A has the same length as A but points the opposite way;
• −1.5A points opposite to A and is 1.5 times as long.

The magnitude of a vector is never negative, so for a negative factor the length is multiplied by the size of the number: |−1.5A| = 1.5A. NCERT’s summary (page 43) says the magnitude of λA is “λ times” that of A for any real λ; for negative λ it should be |λ| times. Section 3.3 itself is correct: it writes the negative number as −λ, with λ positive, so its “λ times |A|” is our |λ|A.

For non-zero A and λ, B = λA is parallel to A if λ > 0 and antiparallel if λ < 0. Drawn from a common tail, they lie on the same line. In components (NCERT Section 3.5, page 32), λA = λA_xî + λA_yĵ: every component is multiplied by λ. So two non-zero vectors are parallel or antiparallel exactly when their components are in the same ratio, B_x/A_x = B_y/A_y = λ, and the sign of λ tells you which. Take ratios only for non-zero components of A. If a component of A is zero, the same component of B must also be zero; the component equations B = λA avoid division by zero.

The negative of a vector is −B = (−1)B, and NCERT uses it to define subtraction: A − B = A + (−B) (NCERT Eq. 3.5, page 30).

Multiplying by a quantity with dimensions

The number λ can be a physical scalar with its own dimensions. Then the dimensions of λA are the product of the dimensions of λ and of A (NCERT page 29). Many laws of physics are of this form:

• displacement = constant velocity × elapsed time (NCERT’s example, page 29); for varying velocity, use the average velocity over that interval;
• momentum p = mv: a scalar mass times a velocity;
• force F = ma, from Newton’s second law: for a constant positive mass in an inertial frame, a non-zero net force points along the acceleration.

Rules of scalar multiplication

For real numbers λ and μ and vectors A and B (beyond the NCERT text, but used throughout the chapter):

• λ(A + B) = λA + λB;
• (λ + μ)A = λA + μA;
• λ(μA) = (λμ)A;
• 1A = A; 0A = 0 and λ0 = 0 (the last two are NCERT Eq. 3.4, page 30).

Dividing a vector by a non-zero number λ means multiplying by 1/λ. In particular, dividing a non-zero vector A by its own magnitude gives a vector of length 1 along A, its unit vector, so A = A n̂ (NCERT Eq. 3.10, page 32). Division by a vector is not defined. Multiplying a vector by a scalar is also not the same as the scalar product of two vectors, which gives a number.

Solved examples

Common mistakes to avoid

Quick revision summary

• λA has magnitude |λ|A; same direction as A if λ > 0, opposite if λ < 0; 0 if λ = 0.
• B = λA means A and B are parallel (λ > 0) or antiparallel (λ < 0).
• Product dimensions multiply: constant velocity × elapsed time = displacement; mass × velocity = momentum. Dimensions can also cancel, as when dividing a vector by its magnitude.
• λ(A + B) = λA + λB; (λ + μ)A = λA + μA.
• A non-zero vector divided by its magnitude gives its unit vector: A = A n̂.

Questions students ask

What is the negative of a vector, and can a vector be negative?

The negative of a vector A is −A = (−1)A, a vector with the same magnitude as A but the opposite direction, so that A + (−A) = 0; if A is 5 m s^⁻¹ towards the north, −A is 5 m s^⁻¹ towards the south. It is the same thing as the opposite vector, and any negative multiple of A, such as −3A, is antiparallel to it. A vector as a whole is neither positive nor negative, but its components can be: −7î + 24ĵ has a negative x-component and a magnitude of 25.

Can a unit vector be negative?

Yes, a unit vector can point in a negative direction, such as −î along the −x-axis, but its magnitude is always 1. Multiplying a unit vector n̂ by −1 reverses it without changing its length, so −n̂ is the unit vector opposite to n̂. Multiplying it by a number λ gives a vector of length |λ|, so the result is a unit vector only for λ = ±1.

Why is momentum a vector quantity?

Momentum is a vector because it is a positive scalar, the mass, multiplied by a vector, the velocity: p = mv has magnitude mv and always points in the direction of motion. A 0.5 kg ball moving at 4 m s^⁻¹ towards the east has a momentum of 2 kg m s^⁻¹ towards the east; moving west at the same speed, it has a different momentum, 2 kg m s^⁻¹ towards the west. Its kinetic energy, ½mv^² = 4 J, is the same in both cases, because kinetic energy depends only on the speed and is a scalar.

What are the different ways of multiplying vectors?

Three vector multiplication operations used in this course are: by a scalar, and with each other as the scalar product or the vector product. Multiplying a vector by a scalar λ gives a vector λA along the same line. The scalar (dot) product, A·B = AB cos θ, is a number and gives work, while the vector (cross) product, A × B, is a vector of magnitude AB sin θ perpendicular to both and gives torque.

Practice questions

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

1. Q1. A force F of 10 N acts vertically upwards. Describe the force −0.5F.

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   5 N, vertically downwards: half the size, opposite direction.

2. Q2. What is the magnitude of −3A if A = 2 units?

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   |−3|A = 3 × 2 = 6 units.

3. Q3. A car’s acceleration is 2 m s^⁻² towards the south and its mass is 1000 kg. In which direction is the net force on it, and how large is it?

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   F = ma = 1000 kg × 2 m s^⁻² = 2000 N, towards the south: a positive mass keeps the direction of the acceleration.