Problem 36 - Master Resource Book in JEE Main Physics

Question

If \(\mathbf{P}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{Q}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}\), then \(\mathbf{P} \cdot \mathbf{Q}\) is (a) zero (b) 6 (c) 12 (d) 15

Step-by-Step Solution

Answer

(c) 12

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}} \) and \( \mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + b_3 \hat{\mathbf{k}} \) is given by \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \). This computes the sum of the products of their corresponding components.

Step 2: Identify Components of Vectors P and Q

Note the components of given vectors: \( \mathbf{P} = 2 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + 1 \hat{\mathbf{k}} \). This means \( a_1 = 2 \), \( a_2 = -3 \), and \( a_3 = 1 \). For \( \mathbf{Q} = 3 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \), we have \( b_1 = 3 \), \( b_2 = -2 \), and \( b_3 = 0 \).

Step 3: Apply the Dot Product Formula

Substitute the components of \( \mathbf{P} \) and \( \mathbf{Q} \) into the dot product formula: \( \mathbf{P} \cdot \mathbf{Q} = (2)(3) + (-3)(-2) + (1)(0) \).

Step 4: Perform the Multiplications

Calculate each product: \( (2)(3) = 6 \), \( (-3)(-2) = 6 \), and \( (1)(0) = 0 \).

Step 5: Add the Products

Add the results of the multiplications: \( 6 + 6 + 0 = 12 \).

Step 6: Select the Correct Option

The value of the dot product is 12, which corresponds to option (c).

Key Concepts

Vector Multiplication

Vector multiplication can be a bit tricky because there are two primary ways to multiply vectors: the dot product and the cross product. Here, we'll focus on the dot product. The dot product is a way to multiply two vectors, producing a single scalar quantity rather than a vector. This makes it different from vector addition or cross-product multiplication.In simple terms, the dot product tells us how much one vector extends in the direction of another. It's useful for determining angles between vectors and for projections. The formula for the dot product is:\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]Where:

• \(a_1, a_2, a_3\) are the components of vector \(\mathbf{A}\)
• \(b_1, b_2, b_3\) are the components of vector \(\mathbf{B}\)

The result, as mentioned, is a scalar. This means when you apply the dot product to two three-dimensional vectors, you get a number, not another vector.

Vector Components

Understanding vector components is essential when dealing with vectors in physics or mathematics. Each vector in three-dimensional space can be broken down into three parts: its components along the x, y, and z axes. These components tell us the magnitude of the vector's projection on each axis.For example, if you have a vector \( \mathbf{P} = 2 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \), it means:

• \(2\) is the component along the x-axis (\(\hat{\mathbf{i}}\))
• -3 is the component along the y-axis (\(\hat{\mathbf{j}}\))
• 1 is the component along the z-axis (\(\hat{\mathbf{k}}\))

Each component represents how far and in which direction the vector points along that axis. Recognizing these components allows us to perform operations like the dot product because we can see how each part interacts with those of another vector.

Dot Product Calculation

The calculation of the dot product might seem complex, but it boils down to simple arithmetic.To calculate the dot product for vectors \(\mathbf{P}\) and \(\mathbf{Q}\), substitute each component into the dot product formula:\[ \mathbf{P} \cdot \mathbf{Q} = (a_1)(b_1) + (a_2)(b_2) + (a_3)(b_3) \]For this given exercise:

• \( \mathbf{P} = 2 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + \hat{\mathbf{k}} \)
• \( \mathbf{Q} = 3 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + 0 \hat{\mathbf{k}} \)

The components are:

• \(a_1 = 2, a_2 = -3, a_3 = 1\)
• \(b_1 = 3, b_2 = -2, b_3 = 0\)

Perform the arithmetic:

• \((2)(3) = 6\)
• \((-3)(-2) = 6\)
• \((1)(0) = 0\)

Finally, add these results together:\[ 6 + 6 + 0 = 12 \]Thus, the dot product of vectors \(\mathbf{P}\) and \(\mathbf{Q}\) is 12, clearly showing that despite their different orientations, their combined influence results in a scalar value of 12.