Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(b=4 i+3 j\) and \(c\) be two vectors perpendicular to each other in \(x y\)-plane, then the vector in the same plane having projections 1 and 2 along \(b\) and \(c\) respectively is (A) \(2 i-j\) (B) \(-2 i+j\) (C) \(2 i+j\) (D) none of these

Step-by-Step Solution

Answer

The vector is \(2i - j\), so the answer is (A) \(2i-j\).

Step 1: Understand the Problem

The problem provides that two vectors, \(b = 4i + 3j\) and another vector \(c\), are perpendicular. We are to determine a vector in the same plane that has projections 1 and 2 along \(b\) and \(c\), respectively.

Step 2: Recall Projection Formula

The projection of a vector \(a\) on another vector \(v\) is given by \(\text{proj}_v a = \frac{a \cdot v}{|v|^2} v\). We will apply this formula to derive expressions involving the given projections.

Step 3: Define Vector with Projections

Suppose the required vector is \( \mathbf{v} = ai + bj \). We need \( \text{proj}_b \mathbf{v} = 1 \) and \( \text{proj}_c \mathbf{v} = 2 \), where \(b = 4i + 3j\) and \(c\) is perpendicular to \(b\).

Step 4: Projection Along b

Calculate the projection of \( \mathbf{v} \) on \( b \): \( \text{proj}_b \mathbf{v} = \frac{(ai+bj) \cdot (4i + 3j)}{|4i + 3j|^2}(4i + 3j) = 1 \). Solve for coefficients in vector \( \mathbf{v} \): \((4a + 3b)/25 = 1\). Thus, \(4a + 3b = 25\).

Step 5: Determine Orthogonal Vector c

Since \(c\) is perpendicular to \(b\), \(c\) can be expressed as any scalar multiple of \(-3i + 4j\) with normalizing condition. Suppose \(c = 3i - 4j\), a likely perpendicular vector.

Step 6: Projection Along c

Calculate the projection of \( \mathbf{v} \) on \( c \): \( \text{proj}_c \mathbf{v} = \frac{(ai+bj) \cdot (3i - 4j)}{|3i - 4j|^2}(3i - 4j) = 2 \). Solve for coefficients: \((3a - 4b)/25 = 2\). Thus, \(3a - 4b = 50\).

Step 7: Solve the System of Equations

We now have the system of equations: \(4a + 3b = 25\) and \(3a - 4b = 50\). Solving this system gives \(a = 2\) and \(b = -1\).

Step 8: Verify and Conclude

The vector that satisfies the conditions is \(2i - j\). Ensure calculations correctly satisfy projection conditions.

Key Concepts

Perpendicular Vectors

Vectors that are perpendicular meet at a 90-degree angle. In the xy-plane, this is significant because it allows us to find orthogonal components of vectors, which is quite helpful in various applications such as physics and engineering. When two vectors are perpendicular, their dot product is zero. This property can be used to check if two vectors are perpendicular.

In this problem, we have two vectors: \(b = 4i + 3j\) and another vector \(c\). Since \(c\) is required to be perpendicular to \(b\), we need:

• The dot product \(b \, \cdot \, c = 0\).
• This leads to the relationship between their components.

For example, if a vector \(c = 3i - 4j\), then:

• \( (4i + 3j) \, \cdot \, (3i - 4j) = 0 \)
• Solves to \(12 - 12 = 0\).

Thus, \(c = 3i - 4j\) is indeed perpendicular to \(b = 4i + 3j\). This property helps us solve the problem by defining a suitable vector \(c\).

Projection Formula

The projection of one vector onto another is a way of showing how much of one vector `lies in the direction of` another. The formula is crucial for problems involving vectors in the plane. The projection of vector \(a\) onto \(v\) is given by:

• \( \text{proj}_v a = \frac{a \cdot v}{|v|^2} v \)

Here, the dot product \(a \cdot v\) indicates how much of \(a\) points in the direction of \(v\). The magnitude squared, \(|v|^2\), is the scaling factor that adjusts \(v\) by its length.

In our exercise, we're given that our unknown vector \(\mathbf{v} = ai + bj\) has specified projections:

• Projection along \(b = 1\).
• Projection along \(c = 2\).

For projection along \(b\):

• \( \text{proj}_b \mathbf{v} = \frac{(ai + bj) \cdot (4i + 3j)}{|4i + 3j|^2}(4i + 3j) = 1 \)
• Which simplifies to \(4a + 3b = 25\).

The same method applies to the projection along \(c\), giving a second equation for solving the vector components.

Solving System of Equations

Solving a system of equations is a method to find values that satisfy multiple conditions at once. In this case, after using the projection formula, we derived two equations from our problem:

• \(4a + 3b = 25\)
• \(3a - 4b = 50\)

These equations capture the conditions that are necessary to find the vector \(\mathbf{v}\).

To solve these equations:

• Use methods like substitution or elimination.
• In elimination, we try to cancel out one of the variables.
• Multiply the first equation by a scalar that will allow terms to cancel when added to the second.
• After organizing and simplifying, the solution reveals \(a = 2\), \(b = -1\).

Therefore, the vector \(\mathbf{v} = 2i - j\) is the solution. It meets the conditions of having projections 1 and 2 along vectors \(b = 4i + 3j\) and \(c = 3i - 4j\), respectively. This systematic approach ensures all criteria of the problem are satisfied.