If , , and are the unit vectors such that and , then

Both , , and , , are coplanar Explanation: 1. Analyze the Dot Product of Cross Products: The maximum value of the dot product of two vectors is the product of their magnitudes. For any two vectors and : Here, and . The magnitude of a cross product of unit vectors is: Given , the only way this is possible is if: (angle is ) (angle is ) The vectors and must be parallel and in the same direction. 2. Use the Lagrange Identity: The identity for is: Substitute the given values: 3. Geometrical Interpretation: Since and are equal unit vectors (let's call this normal vector ), lie in the same plane as (the plane perpendicular to ). In this 2D plane: Let be at . Since , the angle between and is . Since , is at . Since , is at (to maintain the direction of the cross product). 4. Calculate other dot products: Angle between and is . Angle betwe

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CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

If $a$ , $b$ , $c$ and $d$ are the unit vectors such that $(a \times b) \cdot (c \times d) = 1$ and $(a \cdot c) = \frac{1}{2}$ , then

Choose the correct answer:

Explanation

1. Analyze the Dot Product of Cross Products:

The maximum value of the dot product of two vectors is the product of their magnitudes.

For any two vectors $\vec{u}$ and $\vec{v}$ :

\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \phi \leq |\vec{u}| |\vec{v}|

Here, $\vec{u} = \vec{a} \times \vec{b}$ and $\vec{v} = \vec{c} \times \vec{d}$ .

The magnitude of a cross product of unit vectors is:

|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin \theta_1 = \sin \theta_1 \leq 1

|\vec{c} \times \vec{d}| = |\vec{c}||\vec{d}| \sin \theta_2 = \sin \theta_2 \leq 1

Given $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1$ , the only way this is possible is if:

$|\vec{a} \times \vec{b}| = 1 \implies \vec{a} \perp \vec{b}$ (angle is $90^\circ$ )
$|\vec{c} \times \vec{d}| = 1 \implies \vec{c} \perp \vec{d}$ (angle is $90^\circ$ )
The vectors $(\vec{a} \times \vec{b})$ and $(\vec{c} \times \vec{d})$ must be parallel and in the same direction.

2. Use the Lagrange Identity:

The identity for $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})$ is:

(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})

Substitute the given values:

1 = \left(\frac{1}{2}\right)(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})

3. Geometrical Interpretation:

Since $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are equal unit vectors (let's call this normal vector $\hat{n}$ ), $\vec{a}, \vec{b}$ lie in the same plane as $\vec{c}, \vec{d}$ (the plane perpendicular to $\hat{n}$ ).

In this 2D plane:

Let $\vec{a}$ be at $0^\circ$ .

Since $\vec{a} \cdot \vec{c} = \frac{1}{2}$ , the angle between $\vec{a}$ and $\vec{c}$ is $60^\circ$ .

Since $\vec{a} \perp \vec{b}$ , $\vec{b}$ is at $90^\circ$ .

Since $\vec{c} \perp \vec{d}$ , $\vec{d}$ is at $60^\circ + 90^\circ = 150^\circ$ (to maintain the direction of the cross product).

4. Calculate other dot products:

Angle between $\vec{b}$ and $\vec{d}$ is $|150^\circ - 90^\circ| = 60^\circ$ .

$\vec{b} \cdot \vec{d} = \cos(60^\circ) = \frac{1}{2}$
Angle between $\vec{a}$ and $\vec{d}$ is $150^\circ$ .

$\vec{a} \cdot \vec{d} = \cos(150^\circ) = -\frac{\sqrt{3}}{2}$
Angle between $\vec{b}$ and $\vec{c}$ is $|90^\circ - 60^\circ| = 30^\circ$ .

$\vec{b} \cdot \vec{c} = \cos(30^\circ) = \frac{\sqrt{3}}{2}$

5. Verify with the identity:

(\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c}) = \left(\frac{1}{2}\right)\left(\frac{1}{2}\right) - \left(-\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)

= \frac{1}{4} + \frac{3}{4} = 1

Final Conclusion:

Under these conditions, $\vec{b}$ and $\vec{d}$ are non-parallel, and $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are coplanar. Specifically, $\vec{b} \cdot \vec{d} = \frac{1}{2}$ and the vectors satisfy the geometric arrangement described above.

(Correct Answer)