If |a|=5, |b|=4, |c|=3 thus what will be the value of |a.b+b.c+c.a|, given that a+b+c=0 (a) 25 (b) 5

Q: Wait, why do we square the vector sum? How does that help?

Great question! Squaring a vector sum is a powerful technique because it transforms the vector equation into a scalar one. Specifically, the dot product of a vector with itself is the square of its magnitude: $\vec{v}\cdot\vec{v} = |\vec{v}|^2$. By expanding $(\vec{a}+\vec{b}+\vec{c})\cdot(\vec{a}+\vec{b}+\vec{c})$, we naturally generate the dot product terms $\vec{a}\cdot\vec{b}$, $\vec{b}\cdot\vec{c}$, and $\vec{c}\cdot\vec{a}$ that the question asks for.

Q: Why is the final answer positive if the dot product sum is negative?

The question specifically asks for the absolute value (or modulus) of the sum, denoted by $|\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}|$. While the sum itself is $-25$, the modulus operator ensures the final result is the positive magnitude, which is $25$.

Q: Can I solve this by finding the angles between the vectors?

You could, but it would be much harder! You would need to use the Law of Cosines to find the angles between the vectors, then calculate each dot product individually using $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos(\theta)$. The algebraic identity method is much more elegant and less prone to calculation errors.

Question

If $|\vec{a}|=5, |\vec{b}|=4, |\vec{c}|=3$ thus what will be the value of $|\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}|$, given that $\vec{a}+\vec{b}+\vec{c}=0$

Options:

(A) 25

(B) 50

(C) -25

(D) -50

Sigma Percentile · Accepted Answer

### The Geometry of Equilibrium

Imagine you are standing in a vast, open field, holding three arrows of different lengths: $5$ units, $4$ units, and $3$ units.

The problem states that if we place these arrows head-to-tail, they form a closed loop, meaning their vector sum is zero: 
$$\vec{a} + \vec{b} + \vec{c} = 0$$

This is the physical heart of our problem. When three vectors sum to zero, they must form a closed triangle, which is a rigid geometric constraint that dictates their relationship.

### The Algebraic Bridge

We need to find the value of $|\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}|$. To bridge the gap between the vector sum and these dot products, we utilize algebraic expansion.

Just as we expand $(x+y+z)^2$ in basic algebra, we expand the square of our vector sum:
$$(\vec{a} + \vec{b} + \vec{c})^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a})$$

This identity is the key that unlocks the entire problem. It connects the magnitudes we know to the dot products we need to find.

### The Calculation

Since we know $\vec{a} + \vec{b} + \vec{c} = 0$, it follows that the square of this sum must also be zero:
$$(\vec{a} + \vec{b} + \vec{c})^2 = 0$$

Substituting this into our identity, we obtain:
$$0 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a})$$

Now, we plug in the magnitudes provided: $|\vec{a}|=5$, $|\vec{b}|=4$, and $|\vec{c}|=3$. Squaring these yields $25$, $16$, and $9$, which sum to $50$.

Our equation now reads:
$$0 = 50 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a})$$

### The Final Reveal

By transposing $50$ to the other side and dividing by $2$, we find the sum of the dot products:
$$\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -25$$

The negative sign indicates that the vectors are oriented such that the angles between them are obtuse.

Finally, the question asks for the absolute value of this sum. Taking the modulus of $-25$ gives us our final result:
**$25$**

Animated Solution for Mathematics - Vector Algebra: If $∣ a ∣ = 5, ∣ b ∣ = 4, ∣ c ∣ = 3$ thus what will be the value of $∣ a \cdot b + b \cdot c + c \cdot a ∣$ , given that $a + b + c = 0$

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Visualized Solution

Vector Magnitudes

Closed Triangle Condition

The Vector Identity

Applying $a + b + c = 0$

Substituting Magnitudes

Calculating the Squares

Summing the Constants

Transposing the Sum

Isolating the Dot Product Sum

Applying the Modulus

The Sigma Insight: Scalar (Dot) Product

The Geometry of Equilibrium

The Algebraic Bridge

The Calculation

The Final Reveal

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Animated Solution for Mathematics - Vector Algebra: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣a∣=5,∣b∣=4,∣c∣=3 thus what will be the value of ∣a⋅b+b⋅c+c⋅a∣, given that a+b+c=0