Q.Find the sum of the vectors a=i^−2j^+k^, b=−2i^+4j^+5k^ and c=i^−6j^−7k^.
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Triangle Law of Vector Addition
How do you combine two vectors into a single one? If you make two journeys one after the other, the net journey is a single vector from where you started to where you finished. That is exactly the triangle law.
The law
If two vectors are represented, in magnitude and direction, by two sides of a triangle taken in order (the tip of the first joined to the tail of the second), then their sum is represented by the third side taken in the reverse order — from the tail of the first to the tip of the second.
Place a, then start b where a ends. The arrow that closes the triangle, drawn from the start of a to the end of b, is the resultant a+b.
AB+BC=AC
Why it works
Read the vectors as directed displacements: going from A to B and then B to C lands you at C, and the single displacement that achieves the same is A to C. The intermediate point B cancels — only the overall start and finish survive.
Consequences
- Commutative: a+b=b+a. Completing the triangle the other way gives the same closing side — which is why the parallelogram law agrees with the triangle law.
- Closed triangle = zero: if three vectors form a triangle taken in order, AB+BC+CA=0, since you return to the start.
- To subtract, add the negative: a−b=a+(−b), reversing b before joining it. …
Concept: Vector Addition — add corresponding components.
Step 1: Write the vectors in component form:
a=(1,−2,1), b=(−2,4,5), c=(1,−6,−7).
Step 2: Add the i^ components: 1+(−2)+1=0.
Step 3: Add the j^ components: −2+4+(−6)=−4. …
Vector addition is done component-wise: add the i^, j^, and k^ coefficients separately. The sum is − 4j^−k^.
The idea is simple: when you add vectors, you combine their effects along each direction independently. Think of it like adding apples to apples and oranges to oranges — the i^ parts only combine with other i^ parts, and so on. This works because the unit vectors i^,j^,k^ are mutually perpendicular and form a basis for 3D space.
Let’s go through it step by step.
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Identify the components of each vector.
a=1i^−2j^+1k^
b=−2i^+4j^+5k^
c=1i^−6j^−7k^
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Add the i^-components.
1+(−2)+1=0
So the i^-component of the sum is 0i^ — it cancels out completely.
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Add the j^-components.
−2+4+(−6)=−4
So the j^-component is −4j^.
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Add the k^-components.
1+5+(−7)=−1
So the k^-component is −k^.
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Write the resultant vector. …
Method: Adding vectors component-wise
Use this whenever you must sum two or more vectors given in i^,j^,k^ form.
Steps
Step 1: Line up like components
Because i^,j^,k^ are mutually perpendicular basis directions, each combines only with its own kind — "i^ with i^", and so on.
Step 2: Add each component separately
∑v=(∑x)i^+(∑y)j^+(∑z)k^ …
Common Mistakes
Mistake 1: Sign errors when adding negative components
Why it's wrong: the j^ sum is −2+4−6=−4 and the k^ sum is 1+5−7=−1; a dropped minus gives a wrong resultant. Correct approach: add each column carefully with signs, giving 0i^−4j^−k^.
Mistake 2: Mixing components across directions …
- CBSE 2026Set ANNUAL1 markMCQQ.The sum of the vectors a=i^−2j^+k^, b=−2i^+4j^+5k^ and c=i^−6j^−7k^ is(a) −4j^−k^(b) 4i^−k^(c) 4j^+5k^(d) i^+4j^−k^
›Reveal solutionSolution
Add the i^,j^,k^ components of the three vectors separately.
a+b+c:
i^: 1−2+1=0
j^: −2+4−6=−4
k^: 1+5−7=−1
…
- CBSE 20251 markMCQQ.If a+b+c=0, ∣a∣=37, ∣b∣=3 and ∣c∣=4, then angle between b and c is (A) 6π (B) 4π (C) 3π (D) 2π
›Reveal solutionSolution
Using the triangle law of vector addition, the three vectors form a closed triangle. Applying the cosine rule to the triangle formed by b and c (with a as their resultant) gives cosθ=21, so the angle between b and c is 3π.
The key insight here is that when three vectors add to zero, they form the sides of a triangle taken head-to-tail. This is the Triangle Law of Vector Addition in reverse: if a+b+c=0, then a+b=−c, meaning the sum of any two gives the negative of the third. Geometrically, the three vectors can be arranged as three sides of a triangle, with each side representing one vector's magnitude and direction.
So we have a triangle whose sides have lengths ∣a∣=37, ∣b∣=3, and ∣c∣=4. The angle between b and c is the interior angle of this triangle at the vertex where b and c meet. In the triangle, the side opposite this angle is a (since a connects the tail of b to the head of c when arranged head-to-tail).
Now we apply the cosine rule from trigonometry: in any triangle with sides p, q, r, where r is opposite the angle θ between p and q, we have r2=p2+q2−2pqcosθ.
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Identify the sides: Let the angle between b and c be θ. Then the side opposite θ is ∣a∣=37. The two sides forming the angle are ∣b∣=3 and ∣c∣=4.
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Write the cosine rule:
∣a∣2=∣b∣2+∣c∣2−2∣b∣∣c∣cosθ
- Substitute the given magnitudes:
(37)2=32+42−2(3)(4)cosθ
37=9+16−24cosθ
37=25−24cosθ
- Solve for cosθ:
37−25=−24cosθ
12=−24cosθ
cosθ=−2412=−21
Watch outA common mistake is to forget the minus sign in the cosine rule. The formula is r2=p2+q2−2pqcosθ, not +2pqcosθ. Also, note that cosθ came out negative here — that's fine; it just means the angle is obtuse. But wait — let's check: cosθ=−21 gives θ=32π, which is not among the options. Something is off.
Let's re-examine the geometry. The angle between b and c in the vector equation is not the interior angle of the triangle where they meet head-to-tail. When vectors are placed head-to-tail, the angle between b and c is actually the exterior angle at that vertex, because c starts at the head of b, so the direction of c is away from b's head. The interior angle of the triangle is the supplement of the angle between the vectors. …
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- CBSE 2025Set A1 markQ.The vector sum of the three sides of a triangle taken in order is ______.
›Reveal solutionSolution
Walking along all three sides of a triangle in order brings you back to the starting point, so the net vector displacement is zero.
Let the triangle have vertices A, B, C. Taking the sides in order as vectors: AB+BC+CA.
…
- CBSE 2025Set ANNUAL1 markQ.Write Associative property for addition of any three vectors a,b and c.
›Reveal solutionSolution
Vector addition is associative — the way three vectors are grouped while adding does not affect the result.
For any three vectors a,b,c, the associative property of vector addition states:
(a+b)+c=a+(b+c)
…
- CBSE 2023Set A1 markQ.The vector sum of the three sides of a triangle taken in order is ______.
›Reveal solutionSolution
Traversing a triangle's three sides in order returns you to the starting point, so the vector sum is zero.
Let the triangle have vertices A,B,C. Taking the sides in order as AB,BC,CA:
AB+BC+CA=AC+CA=AC−AC=0. …
- CBSE 2022Set ANNUAL1 markMCQQ.If a=2i^+3k^, b=i^+2j^−3k^ and c=3j^−4k^, then determine a−b+2c.(a) 2i^−4j^+8k^(b) i^−4j^−8k^(c) i^+4j^−8k^(d) i^+4j^+8k^
›Reveal solutionSolution
Add/subtract the vectors component-by-component (i^,j^,k^ separately).
Given a=2i^+0j^+3k^, b=i^+2j^−3k^, c=0i^+3j^−4k^.
a−b=(2−1)i^+(0−2)j^+(3−(−3))k^=i^−2j^+6k^.
2c=6j^−8k^.
a−b+2c=(1+0)i^+(−2+6)j^+(6−8)k^=i^+4j^−2k^.
…
- CBSE 2020Set HE8231 markQ.Write true or false: The vector sum of the three sides of a triangle taken in order is 0.
›Reveal solutionSolution
The statement is True — this is a direct consequence of the triangle law of vector addition.
Let the triangle have vertices A,B,C. Taking the sides in order means AB,BC,CA (each side's vector starts where the previous one ended).
By the triangle law, AB+BC=AC.
Adding CA to both sides: …
- CBSE 2018Set ANNUAL1 markMCQQ.If two forces of 3 units and 4 units are acting at an angle 90°, then its resultant force will be:(a) 3 units(b) 4 units(c) 5 units(d) 0 unit
›Reveal solutionSolution
Two forces at right angles combine like the legs of a right triangle — the resultant is the hypotenuse, found with the Pythagorean form of the parallelogram law.
When two forces (or vectors) P and Q act at a point with an angle θ between them, the magnitude of their resultant is:
R=P2+Q2+2PQcosθ
…
- CBSE 2016Set ANNUAL1 markQ.State the triangle law of vectors.
›Reveal solutionSolution
definition recall
Triangle law of vectors: If two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant (sum) is represented in magnitude and direction by the third side of the triangle taken in the reverse order.
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