Vector Addition Questions

Using vector addition, two vectors, a and b, can be added together, and a + b will be the resultant vector.

Here are some conditions that are to be followed while adding vectors.

• Only the same nature of vectors can be added together. For example, acceleration must be added to acceleration and not mass.
• Vectors and scalars cannot be added.

Consider two vectors C and D. Where, C = Cxi + Cyj + Czk and D = Dxi + Dyj + Dzk. Then, the resultant vector (or vector sum) R = C + D = (Cx + Dx)i + (Cy + Dy)j + (Cz + Cz) k

Read more about vectors and scalars: Vectors and Scalars

Vectors can be divided into vertical and horizontal components. For example, a vector A at an angle Φ, as shown in the below-given image, can be divided into its horizontal and vertical components as:

Therefore, vector A can be expressed as:

A = Ax + Ay

Ax = A cos Φ

Ay = A sin Φ

Triangle Law of Addition of Vectors

According to this law, two vectors should be placed so that the head of the first vector links the tail of the second vector, and by placing them this way, both vectors can be added together.

Here are the steps to perform the addition of vectors using the triangle law –

• Firstly, the two vectors M and N are positioned together so that the head of vector M joins the tail of vector N.
• And then, a resultant vector S is drawn so that it connects the =of M to the head of N to find the sum.
• Thus, mathematically, the sum, or the resultant, vector S, in the below-given image can be articulated as S = M + N.

Parallelogram Law of Addition of Vectors

Parallelogram law of the addition of vectors is another law for the addition of vectors. Let’s take two vectors, p and q, as shown below.

Two adjacent sides of a parallelogram are formed by them in their magnitude and direction. The sum p + q is represented in magnitude and direction by the diagonal of the parallelogram through their mutual point. This is the Parallelogram law of vector addition.

In the above-given figure, using the Triangle law, we can accomplish the following:

OP + PR = OR

OP + OQ = OR, since PR = OQ

Important Questions on Addition of Vectors

1) What is the Vector Addition Rule?

Vector addition rules are the rules to be followed while adding vectors.

1. Only the same nature of vectors can be added. For example, acceleration must be added with acceleration only and not mass.
2. Vectors and scalars cannot be added.

2) Is the Addition of Vectors Commutative?

Yes, vector addition is commutative; for any two random vectors c and d,

c + d = d + c.

3) Differentiate between parallelogram law of vector addition and triangle law of vector addition.

For any two given vectors, the parallelogram law of vector addition says that the diagonal becomes the resultant sum vector,. In contrast, for the same condition the triangle law of vector addition says that, the third side of the triangle will become the resultant sum vector.

4) Find the addition of vectors PQ and QR, where PQ = (3, 2) and QR = (2, 6)

We will perform the vector addition by adding their corresponding components

PQ + QR = (3, 4) + (2, 6)

= (3 + 2, 4 + 6) = (5, 10).

5) What is the Formula of Parallelogram Law of Addition of Vectors?

The magnitude of the sum, |a + b|, is given by √(a²+b²+2abcos(θ)) for two given vectors a and b making an angle θ, according to the parallelogram law of addition of vectors.

6) What is the Associative Property of Addition of Vectors?

For any three random vectors a, b, and c, addition is associative (a + b) + c = a + (b + c) i.e., the order of addition does not matter.

7) If a = <1, -1> and b = <2, 1> then find the unit vector in the direction of addition of vectors a and b.

The vector sum will be written as: a + b = <1, -1> + <2, 1> = <1 + 2, -1 + 1> = <3, 0>

Its magnitude will be, |a + b| = √ (3² + 0²) = 3.

In the direction of vector addition, the unit vector is: (a + b) / |a + b| = <3, 0> / 3 = <1, 0>. The required unit vector is, <1, 0>.

8) What will be the magnitude of the resulting vector when two vectors in the same direction are added?

When two vectors in the same direction are added to each other, then, the lengths will be added. The resultant vector will bear the resultant length where length is the magnitude of the vector. Hence, the magnitudes add to give the magnitude of the resultant vector.

9) A vector, 6 units from the origin, along the X axis, is added to the vector 12 units from the origin along the Y axis. What is the resultant vector?

The vector 6 units from the origin and along the X axis is 6î. The vector 12 units from the origin and along Y axis is 12ĵ. Hence, the sum is 6î + 12ĵ.

10) Adding 3î + 8ĵ and î + ĵ gives

When 3î + 8ĵ is added to î + ĵ, the corresponding components get added. Hence, the answer is 4î + 9ĵ.

Practice Questions

1. The vector 45î + 32ĵ is added to a vector. The result gives 15î + 3ĵ as the answer. The unknown vector is:
2. A vector, 7 units from the origin, along the X axis, is added to vector 3 units along the Y axis. What is the resultant vector?
3. What is the angle between B and C if three vectors A, B and C have magnitudes 6, 12 and 14 and →A+→B=→CA→+B→=C→ ?
4. 5N and 12 N are two force vectors and at what angle should they be added to get a resultant vector of 13 N?
5. What will be the displacement from the initial point, if a person travels 10 km North and 20 km East?

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