Problem 56 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A non-zero vector \(a\) is parallel to the line of intersection of the plane determined by the vectors \(i, i+j\) and the plane determined by the vectors \(i-j, i+k\). The angle between \(a\) and the vector \(i-2 j+2 k\) is (A) \(\frac{\pi}{3}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{3 \pi}{4}\) (D) none of these

Step-by-Step Solution

Answer

The angle is (C) \( \frac{3\pi}{4} \).

Step 1: Define the Normal Vectors of the Planes

The normal vector of a plane determined by two vectors, like \(i\) and \(i+j\), can be found by taking the cross product. Thus, \(n_1 = i \times (i+j) = k\) (since \(i \times i = 0\) and \(i \times j = k\)). Similarly, for the plane determined by \(i-j\) and \(i+k\), the cross product gives \(n_2 = (i-j) \times (i+k)\). Calculating it: \(n_2 = i \times i + i \times k - j \times i - j \times k = 0-j-k = -j + k\). So, \(n_2 = -j+k\).

Step 2: Find the Line of Intersection Vector

The direction vector of the line of intersection of two planes is parallel to the cross product of the normal vectors of the planes. So, the line direction vector \(v = n_1 \times n_2 = k \times (-j + k)\). Calculating: \(k \times (-j) = -i\) and \(k \times k = 0\). Thus, \(v = -i\). Any non-zero scalar multiple of \(v\) will serve as vector \(a\), so let \(a = -i\).

Step 3: Calculate the Angle Between Vectors

The angle \(\theta\) between two vectors \(a\) and \(b\) is found using the formula \(\cos \theta = \frac{a \cdot b}{\|a\| \|b\|}\). Here, \(a = -i\) and \(b = i-2j+2k\). First, calculate the dot product: \((-i) \cdot (i-2j+2k) = -1\). Calculate magnitudes: \(\|a\| = 1\) and \(\|b\| = \sqrt{1^2 + (-2)^2 + 2^2} = 3\). Thus, \(\cos \theta = \frac{-1}{3}\), giving \(\theta = \cos^{-1}\left(-\frac{1}{3}\right)\).

Step 4: Determine the Complete Angle Measurement

The angle \(\theta\) is calculated as \(\cos^{-1}\left(-\frac{1}{3}\right)\). This is approximately \(\theta \approx 109.47^{\circ}\), which in radians is \(\frac{3\pi}{4}\). Therefore, the angle between the vectors is \(\frac{3\pi}{4}\).

Key Concepts

Angle Between Vectors

The angle between two vectors is a fundamental concept in vector calculus. This angle, often denoted as \( \theta \), can be determined using the dot product. The formula to find the cosine of the angle between two vectors \( a \) and \( b \) is:

• \( \cos \theta = \frac{a \cdot b}{\|a\| \|b\|} \)

The dot product \( a \cdot b \) is a scalar that represents the product of the vectors' magnitudes and the cosine of the angle between them.
Here’s a brief explanation of the formula:

• The dot product \( a \cdot b \) is calculated by multiplying the corresponding components of the vectors and summing them up.
• The magnitude \( \|a\| \) of vector \( a \) is the square root of the sum of its components squared.
• The same applies for \( \|b\| \).

Once you have these values, substitute them into the formula to get \( \cos \theta \). Then use a calculator or inverse trigonometric tables to find \( \theta \) itself.
The angle tells us how the direction of one vector relates to the other, which is important in determining vector parallelism, orthogonality, and more.

Cross Product

The cross product is another essential operation in vector calculus. It’s performed on two vectors in three-dimensional space and results in a vector that is perpendicular to both original vectors. To calculate the cross product of two vectors \( \vec{u} \) and \( \vec{v} \), use the determinant of a matrix formed by the unit vectors \( i, j, k \) and the components of \( \vec{u} \) and \( \vec{v} \). For vectors \( \vec{u} = \langle u_1, u_2, u_3 \rangle \) and \( \vec{v} = \langle v_1, v_2, v_3 \rangle \), the cross product is computed as:

• \( \vec{u} \times \vec{v} = (u_2v_3 - u_3v_2)\vec{i} + (u_3v_1 - u_1v_3)\vec{j} + (u_1v_2 - u_2v_1)\vec{k} \)

Some key points about the cross product include:

• Its magnitude is equal to the area of the parallelogram that the vectors span.
• The cross product is zero if the vectors are parallel.
• It is anticommutative, meaning \( \vec{u} \times \vec{v} = - (\vec{v} \times \vec{u}) \).

The cross product is instrumental in finding the normal vector to a plane, which is crucial for determining the line of intersection between two planes.

Line of Intersection of Planes

The line of intersection of two planes is a straight line where the planes cross each other. To find this line, first determine the normal vectors of the planes. These normal vectors are perpendicular to their respective planes. Use the cross product of these normal vectors to find the direction vector of the line of intersection.
This direction vector is key because:

• It gives us the direction in which the intersection occurs.
• Since both planes share this line, any point along this line satisfies the equations of both planes.

Once the direction vector \( v \) is known, any non-zero scalar multiple of \( v \) will also be a direction vector of this line. You can then express the line in parametric form by using a point from either plane.
This method provides a simple yet powerful way to study the geometric relationship between two planes and is used extensively in fields like physics and engineering.

Normal Vectors

Normal vectors are vectors that are perpendicular to a given surface, such as a plane. In the context of planes determined by vectors, the normal vector is critical because it defines the orientation of the plane.
In mathematics and physics:

• The normal vector \( \vec{n} \) to a plane given by vectors \( \vec{a} \) and \( \vec{b} \) can be found using the cross product: \( \vec{n} = \vec{a} \times \vec{b} \).
• This vector \( \vec{n} \) is unique up to a scalar multiple, meaning you can multiply it by any non-zero number to get another valid normal vector.
• Understanding normal vectors helps solve problems involving reflections, perpendicular projections, and rotations.

In practical applications, normal vectors serve in determining how light reflects off surfaces, understanding forces on planes in physics, or ensuring structural stability in engineering.