Triangle Law of Vector Addition Questions

---

Very Short Answers Questions [1 Mark Questions]

Ques. What are the different laws of vector addition?

Ans. When two vectors are pointing in different directions, then the magnitude of the resultant vector can be given by using

• Triangle law of vector addition
• Parallelogram law of vector addition
• Polygon law of vector addition

Ques. What is the vector addition rule?

Ans. The following are the vector addition rules carried out when two vectors are added.

The condition rules are:

1. Only the same nature of vectors can be added. For example, force must be added with force only and not distance.
2. Scalars and vectors together can’t be added.

Ques. What is the triangle law of vector addition?

Ans. According to the triangle law of vector addition, if two vectors are represented both in magnitude and direction by two sides of a triangle taken in the same order, then the resultant of these vectors is represented both in magnitude and direction by the third side of the triangle taken in the opposite order.

Ques. What is the parallelogram law of vector addition?

Ans. According to the parallelogram law of vector addition, if two vectors are represented both in magnitude and direction by two adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through the same point.

Ques. What is the polygon law of vector Addition?

Ans. According to the polygon law of vector addition, if a number of vectors are represented both in magnitude and direction by sides of a polygon taken in the same order, then the resultant vector is represented both in magnitude and direction by the closing side of the polygon taken in the opposite order.

---

Short Answers Questions [2 Marks Questions]

Ques. What are the properties of vector addition?

Ans. The three important properties of vector addition are

• Vector addition follows commutative law i.e. (A + B) = (B + A)
• Vector addition is associative i.e. (A + B) + C = A + (B + C)
• Vector addition is distributive i.e λ (A + B) = λA + λB

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

Ans. The basic difference between the parallelogram law and triangle law of vector addition is if two vectors are represented by two sides of a triangle then the third side of the triangle represents the magnitude of the resultant vector, while in parallelogram law of vector addition, the resultant sum vector is given by the diagonal of a parallelogram.

Ques. Calculate the magnitude of the resultant vector; if two vectors P and Q of magnitude 4 units and 6 units respectively, make an angle of 60°

Ans. Given

• The magnitude of vector P = 4 units
• The magnitude of vector Q = 6 units
• The angle between the vectors P and Q, θ = 60°

Using the formula of magnitude resultant vector from the triangle law of vector addition, we get

R = √(P² + Q² + 2AB cos θ)

⇒ R = √(4² + 6² + 2 x 4 x 6 cos 60°)

⇒ R = √(16 + 36 + 48 x 1/2)

⇒ R = √(40 + 12)

⇒ R = √(52)

⇒ R = 7.2 units

Hence, the magnitude of the resultant vector is 7.2 units.

Ques. When two vectors in the same direction are added, what will be the magnitude of the resulting vector?

Ans. Let two vectors of magnitude A and B be in the same direction.

When the two vectors are in the same direction, then the angle between these two vectors is 0°.

Using the formula of magnitude resultant vector from the triangle law of vector addition, we get

R = √(A² + B² + 2AB cos θ)

⇒ R = √(A² + B² + 2AB cos 0)

⇒ R = √(A² + B² + 2AB)

⇒ R = √(A + B)² 

⇒ R = A + B

Hence, when two vectors in the same direction are added, then the magnitude of the resultant vector will be the sum of the magnitudes of each vector.

Read More:

---

Long Answers Questions [3 Marks Questions]

Ques. Two vectors with magnitudes of 2 units and √2 units act on a body. The resultant vector has a magnitude of √10 units. Find the angle between the two given vectors.

Ans. Let the vectors be A and B and their resultant be R.

Given

• A = 2 units
• B = √2 units
• R = √10 units

Using the formula of magnitude resultant vector from the triangle law of vector addition, we get

R = √(A² + B² + 2AB cos θ)

⇒ √10 = √[2² + (√2)² + 2 × 2 × √2 cos θ]

⇒ 10 = 4 + 2 + 4√2 cos θ

⇒ 4√2 cos θ = 10 - 4 - 2

⇒ cos θ = 4/4√2

⇒ cos θ = 1/√2

⇒ θ = 45°

Therefore, the angle between the two given vectors is 45°

Ques. What will be the direction and magnitude of the resulting vector, if P and Q are two vectors making an angle of 45° with each other, and have scales (magnitudes) of 5 and 10 units, respectively?

Ans. Given

• The magnitude of vector P = 5 units
• The magnitude of vector Q = 10 units
• The angle between the vectors P and Q, θ = 45°

Using the formula of magnitude resultant vector from the triangle law of vector addition, we get

R = √(P² + Q² + 2PQ cos θ)

⇒ R = √(5² + 10² + 2 x 5 x 10 cos 45°)

⇒ R = √(25 + 100 + 100 x 1/√2)

⇒ R = √(125+ 50√2)

⇒ R = 13.989 units

The direction of the resultant vector is given by

ϕ = tan^-1 [(Q sin θ)/(P + Q cos θ)]

⇒ ϕ = tan^-1 [(10 x 1/√2)/(5 + 10 x 1/√2)]

⇒ ϕ = tan^-1 [5√2/(5 + 5√2)]

⇒ ϕ = 30.35°

Hence, the direction and magnitude of the resulting vector are 30.35° and 13.989 units respectively.

Ques. Find the angle between the two vectors A and B acting on a body, if the resulting vector has a value of √20 units and their magnitudes are 4 and √4 units.

Ans. Given

• The magnitude of vector A = 4 units
• The magnitude of vector B = √4 units
• The magnitude of the resultant vector of vectors A and B, R = √20 units

Let θ be the angle between the vectors A and B, then using the formula of magnitude resultant vector from the triangle law of vector addition, we get

R = √(P² + Q² + 2PQ cos θ)

⇒ √20 = √(4² + (√4)² + 2 x 4 x √4 cos θ)

⇒ 20 = (16 + 4 + 8√4 cos θ)

⇒ 20 = 20 + 8√4 cos θ

⇒ 8√4 cos θ = 0

⇒ cos θ = 0

⇒ θ = 90°

Hence, the angle between vectors A and B is 90°

---

Very Long Answers Questions [5 Marks Questions]

Ques. The resultant of two forces 3P and 2P is R. If the first is doubled, the resultant is also doubled. Then, find the angle between the two forces.

Ans. The formula for the resultant from the triangle law of vector addition is given by

C² = A² + B² + 2AB cosθ

Case: 1 Given, the resultant of two forces 3P and 2P is R, then

R² = (3P)² + (2P)² + 2 (3P)(2P) cosθ

⇒ R² = 9P² + 4P² + 12P² cosθ

⇒ R² = 13P² + 12P² cosθ …..(i)

Case:2 Given, if the first vector is doubled then the resultant is also doubled, then

(2R)² = (6P)² + (2P)² + 2 (6P)(2P) cosθ

⇒ 4R² = 36P² + 4P² + 24P² cosθ

⇒ 4R² = 40P² + 24P² cosθ

⇒ R² = 10P² + 6P² cosθ …..(ii)

From equation (i) and (ii), we have

13P² + 12P² cosθ = 10P² + 6P² cosθ

⇒ 3P² + 12P² cosθ = 6P² cosθ

⇒ 3P² + 6P² cosθ = 0

⇒ 1 + 2cosθ = 0

⇒ 1 = - 2 cosθ

⇒ cosθ = - 1/2

⇒ θ = 120°

Therefore, the angle between the two vectors is 120°

Ques. If \(\vec{a}+\vec{b}=\vec{c}\) and a² + b² = c², then prove that \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.

Ans. Given \(\vec{a}+\vec{b}=\vec{c}\)

\(|\vec{a}|+|\vec{b}|=|\vec{c}|\)

⇒ a² + b² + 2ab cosθ = c² ….(i)

Where θ is the angle between \(\vec{a}\) and \(\vec{b}\)

Also, given that a² + b² = c²

On substituting the above value in equation (i), we get

c² + 2ab cosθ = c²

⇒ 2ab cosθ = 0

⇒ cosθ = 0

⇒ θ = 90°

Therefore, the angle between \(\vec{a}\) and \(\vec{b}\) is 90°, hence both are perpendicular to each other.

Ques. Two vectors A and B have magnitudes of 4 units and 9 units and make an angle of 30° with each other. Find the magnitude and direction of the resultant sum vector using the triangle law of vector addition formula.

Ans. The formula for the resultant vector from the triangle law of vector addition is given by

|R| = √(A² + B² + 2AB cos θ)

The direction of the resultant vector with respect to the vector A is given by

ϕ = tan^-1 [(B sin θ)/(A + B cos θ)]

Given

• A = 4 units
• B = 9 units
• θ = 30°

So, we have

|R| = √(A² + B² + 2AB cos θ)

= √(4² + 9² + 2 × 4 × 9 cos 30°)

= √(16 + 81 + 72 × √3/2)

= √(97 + 36√3)

= 12.623 units

The direction of R is given by,

ϕ = tan^-1[(B sin θ)/(A + B cos θ)]

⇒ ϕ = tan^-11[(9 sin 30°)/(4 + 9 cos 30°)]

⇒ ϕ = tan^-1[(9 × 1/2)/(4 + 9 × √3/2)]

⇒ ϕ = tan^-1[(4.5)/(11.8)]

⇒ ϕ = 20.87°

Hence, the magnitude and direction of the resultant sum vector is 12.623 units and 20.87° respectively.

---

Previous Year Questions

1. Lassaigne's test for the detection of nitrogen fails in...[NEET UG 1994]
2. The order of reaction is decided by...[KEAM]
3. The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is...[AIIMS 2007]
4. In how many ways can 12 gentlemen sit around a round table so that three specified gentlemen are always together?...[BITSAT 2018]
5. The v−t graph of a moving object is given in figure. The maximum acceleration is...[Chhattisgarh PMT 2007]
6. The change in the optical rotation of freshly prepared solution of glucose is known as...[AIEEE 2011]
7. The percentage of nitrogen in urea is about​..
8. The most reactive nucleophile among the following is...[AIIMS 2003]
9. Which will undergo deprotonation most readily in basic medium?..[JEE Main 2023]
10. The light reflected by a plane mirror may form a real image...[KCET 2002]

For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates

---

Check-Out: