Let u, v and w be vectors such that u + v + w = 0. If |u| = 3, |v| = 4 and |w| = 5, then u · v + v ·

Q: Wait, why is the answer negative? I thought magnitudes were always positive?

That is a great observation! Remember, the dot product is not a magnitude; it is a scalar value that depends on the cosine of the angle between vectors. A negative dot product simply means the angle between the vectors is obtuse. In a closed triangle, the vectors are arranged head-to-tail, which forces the angles between them (when placed tail-to-tail) to be obtuse, leading to a negative result.

Q: Can I solve this using the Law of Cosines instead?

Absolutely! You could find the angles of the triangle formed by sides $3, 4,$ and $5$ using the Law of Cosines. However, the algebraic identity method is much faster, more elegant, and significantly less prone to calculation errors in a high-pressure exam setting like JEE Advanced.

Q: Why does $(\vec{u} + \vec{v} + \vec{w})^2$ equal zero?

The problem states that $\vec{u} + \vec{v} + \vec{w} = \vec{0}$. When we square a vector, we are essentially taking its dot product with itself. Since the sum is the zero vector, we have $\vec{0} \cdot \vec{0} = 0$. This is the 'master key' that unlocks the entire problem.

Question

Let $\vec{u}, \vec{v}$ and $\vec{w}$ be vectors such that $\vec{u} + \vec{v} + \vec{w} = 0$. If $|\vec{u}| = 3, |\vec{v}| = 4$ and $|\vec{w}| = 5$, then $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}$ is

Options:

(A) $47$

(B) $-25$

(C) $0$

(D) $25$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given three vectors, $\vec{u}, \vec{v},$ and $\vec{w}$, whose sum is the zero vector:
$$\vec{u} + \vec{v} + \vec{w} = \vec{0}$$

Geometrically, this implies that if you walk along these vectors sequentially, you return exactly to your starting point. Thus, these three vectors form a closed triangle.

### The Hidden Triple

We are provided with the magnitudes:
$|\vec{u}| = 3, |\vec{v}| = 4, |\vec{w}| = 5$

These values correspond to the sides of a classic $3-4-5$ right-angled triangle. While this geometric insight is elegant, vector algebra provides a universal toolset that functions independently of the triangle's specific angles.

### The Algebraic Master Key

We are tasked with finding the value of the expression $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}$. To bridge the gap between the sum of vectors and their dot products, we utilize the vector identity for the square of a sum:

$$(\vec{u} + \vec{v} + \vec{w}) \cdot (\vec{u} + \vec{v} + \vec{w}) = |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u})$$

Since we know $\vec{u} + \vec{v} + \vec{w} = \vec{0}$, the left side of the equation becomes the dot product of the zero vector with itself, which is $0$.

### The Final Calculation

Substituting the known magnitudes into our expanded equation, we have:

$$0 = |\vec{u}|^2 + |\vec{v}|^2 + |\vec{w}|^2 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u})$$

Plugging in the values $3, 4,$ and $5$:

$$0 = 3^2 + 4^2 + 5^2 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u})$$

$$0 = 9 + 16 + 25 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u})$$

$$0 = 50 + 2(\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u})$$

Isolating our target expression by subtracting $50$ from both sides and dividing by $2$, we obtain the final result:

$$\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = -25$$

The negative sign is a physical consequence of the obtuse angles formed between the vectors when placed tail-to-tail. You have successfully applied a fundamental technique in vector analysis; keep this identity in your toolkit for future challenges.

Animated Solution for Mathematics - Vector Algebra: Let $u, v$ and $w$ be vectors such that $u + v + w = 0$ . If $∣ u ∣ = 3, ∣ v ∣ = 4$ and $∣ w ∣ = 5$ , then $u \cdot v + v \cdot w + w \cdot u$ is

Select Answer:

Visualized Solution

The Zero Vector Sum

Vector Magnitudes

The Vector Expansion Identity

Applying the Zero Condition

Substituting the Magnitudes

Squaring the Magnitudes

Summing the Constants

Isolating the Unknown Term

Final Division

Conclusion and Takeaways

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Hidden Triple

The Algebraic Master Key

The Final Calculation

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