Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of ΔPQR. Then, the angle ∠QPR is

Q: Why do we specifically define vectors $\vec{PQ}$ and $\vec{PR}$? Why not $\vec{QP}$ and $\vec{QR}$?

The angle we are looking for is $\angle QPR$. In geometry, the middle letter denotes the vertex where the angle is located. Therefore, both vectors must originate from $P$ to correctly measure the opening between the two sides.

Q: Is it possible to solve this using the distance formula and the Law of Cosines?

Absolutely! You could calculate the lengths of all three sides $PQ$, $PR$, and $QR$ using the distance formula and then apply the Law of Cosines. However, that is a much longer, more tedious path. Vectors are the elegant, professional way to handle 3D geometry.

Q: What if the dot product comes out to be negative?

That is a great observation! If the dot product is negative, it simply means the angle $\theta$ is obtuse (greater than $90^\circ$). The math handles this automatically through the cosine function, so trust the algebra!

Question

Let $P(3, 2, 3), Q(4, 6, 2)$ and $R(7, 3, 2)$ be the vertices of $	riangle PQR$. Then, the angle $\angle QPR$ is

Options:

(A) $\frac{\pi}{6}$

(B) $\cos^{-1}(\frac{7}{18})$

(C) $\cos^{-1}(\frac{1}{18})$

(D) $\frac{\pi}{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, three-dimensional room. You have three points floating in the air: $P(3, 2, 3)$, $Q(4, 6, 2)$, and $R(7, 3, 2)$. These points form a triangle, $	riangle PQR$.

Your mission is to find the angle $\angle QPR$. While this might feel intimidating in 3D space, vectors serve as the language of 3D geometry, allowing us to translate physical positions into algebraic power.

### The Vector Strategy

To find the angle at vertex $P$, we define the two sides meeting at that point as vectors. We define vector $\vec{PQ}$ and vector $\vec{PR}$.

Think of these as two arrows starting at $P$ and pointing toward $Q$ and $R$. By doing this, we reduce a complex 3D shape into a simple relationship between two arrows. The angle between these arrows is exactly the angle we need.

### The Dot Product: Our Mathematical Compass

To find the angle $	heta$ between two vectors, we use the dot product formula:

$$\cos 	heta = \frac{\vec{PQ} \cdot \vec{PR}}{|\vec{PQ}| |\vec{PR}|}$$

This formula is a bridge between the coordinates and the angle, relating the algebraic dot product in the numerator to the geometric magnitudes in the denominator.

### The Execution

First, we calculate our vectors by subtracting the coordinates of $P$ from $Q$ and $R$:

$$\vec{PQ} = (4-3)\hat{i} + (6-2)\hat{j} + (2-3)\hat{k} = \hat{i} + 4\hat{j} - \hat{k}$$

$$\vec{PR} = (7-3)\hat{i} + (3-2)\hat{j} + (2-3)\hat{k} = 4\hat{i} + \hat{j} - \hat{k}$$

Next, we compute the dot product:

$$\vec{PQ} \cdot \vec{PR} = (1)(4) + (4)(1) + (-1)(-1) = 4 + 4 + 1 = 9$$

Now, we determine the magnitudes of these vectors:

$$|\vec{PQ}| = \sqrt{1^2 + 4^2 + (-1)^2} = \sqrt{18}$$

$$|\vec{PR}| = \sqrt{4^2 + 1^2 + (-1)^2} = \sqrt{18}$$

### The Final Elegance

Substituting these values into our master equation:

$$\cos 	heta = \frac{9}{\sqrt{18} \cdot \sqrt{18}}$$

Since $\sqrt{18} \cdot \sqrt{18} = 18$, the expression simplifies significantly:

$$\cos 	heta = \frac{9}{18} = \frac{1}{2}$$

We know that $\cos 	heta = \frac{1}{2}$ corresponds to **$	heta = \frac{\pi}{3}$** (or $60^\circ$). You have successfully navigated the 3D space and found the angle, demonstrating the power of vectors in turning the abstract into the concrete.

Animated Solution for Mathematics - Three Dimensional Geometry: Let $P (3, 2, 3), Q (4, 6, 2)$ and $R (7, 3, 2)$ be the vertices of $△ P QR$ . Then, the angle $∠ QP R$ is

Select Answer:

Visualized Solution

Visualizing Triangle $P QR$

Identifying the Tool: Vectors

The Dot Product Formula

Calculating Vector $P Q$

Calculating Vector $P R$

Computing the Dot Product

Calculating Magnitude of $P Q$

Calculating Magnitude of $P R$

Substituting into the Formula

Simplifying the Expression

Final Conclusion

The Sigma Insight: Angle Between Two Lines

Analyzing the Setup

The Vector Strategy

The Dot Product

The Execution

The Final Elegance

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Animated Solution for Mathematics - Three Dimensional Geometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(3,2,3),Q(4,6,2) and R(7,3,2) be the vertices of △PQR. Then, the angle ∠QPR is