Problem 59 - Master Resource Book in JEE Main Physics

Question

For what value of \(a, \mathbf{A}=2 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}}\) will be perpendicular to \(\mathbf{B}=4 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}} ?\) (a) 4 (b) zero (c) 3 (d) 1

Step-by-Step Solution

Answer

The vectors are perpendicular for \(a = \frac{7}{2}\), but this is not a given option.

Step 1: Understand the Condition for Perpendicularity

Two vectors \(\mathbf{A}\) and \(\mathbf{B}\) are perpendicular if their dot product is zero. This is the condition we will use to find the value of \(a\).

Step 2: Write Down the Dot Product

The dot product of \(\mathbf{A}=2 \hat{\mathbf{i}}+a\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{B}=4\hat{\mathbf{i}}-2\hat{\mathbf{j}}-\hat{\mathbf{k}}\) is calculated as follows:\[\mathbf{A} \cdot \mathbf{B} = (2)(4) + (a)(-2) + (1)(-1)\]

Step 3: Simplify the Dot Product Expression

Calculate each term of the dot product:- \(2(4) = 8\)- \(a(-2) = -2a\)- \(1(-1) = -1\)Adding these terms gives the expression:\[8 - 2a - 1\] which simplifies to \[7 - 2a\].

Step 4: Solve for \(a\)

Set the dot product equal to zero for perpendicularity:\[7 - 2a = 0\]Solve for \(a\):\[7 = 2a\]\[a = \frac{7}{2}\]

Step 5: Verify Answer With Choices

The calculated value of \(a\) is \(\frac{7}{2}\), which is not directly listed among the choices provided (4, zero, 3, 1). Thus, none of the given options seem to match the solution.

Key Concepts

Dot Product

To understand how vectors interact and relate in space, the concept of the **dot product** is fundamental. It gauges how much one vector contributes to the direction of another. Calculating the dot product between vectors aids in discerning angles and relationships between them. The dot product is computed as follows:

• Given vectors \( \mathbf{A} = a_1 \hat{\mathbf{i}} + b_1 \hat{\mathbf{j}} + c_1 \hat{\mathbf{k}} \) and \( \mathbf{B} = a_2 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + c_2 \hat{\mathbf{k}} \), the dot product \( \mathbf{A} \cdot \mathbf{B} \) is \( a_1a_2 + b_1b_2 + c_1c_2 \).
• The resulting value is not a vector but a scalar—representing the magnitude of overlap between the two vector directions.
• Identical direction means a larger dot product, while opposite direction yields a negative value.

To use dot products effectively, one must understand that they help in calculating the angle between vectors and determining orthogonality. When the dot product is zero, the vectors are not impacting each other's direction, signifying a right angle.

Perpendicular Vectors

Vectors become particularly interest-worthy when they are perpendicular, meaning they intersect at a right angle (90°). Perpendicularity is a critical aspect, especially in physics and geometry, where force vectors or line segments intersect orthogonally. Here’s how perpendicular vectors relate to the dot product:

• The crucial property: the dot product of two perpendicular vectors is zero. When \( \mathbf{A} \cdot \mathbf{B} = 0 \), vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular.
• The analysis of such vectors is useful when exploring angles or resultant vectors in practical scenarios.
• Worked out in real-world applications like electrical engineering, computer graphics, and physics problems.

Perpendicular vectors indicate mutual independence in the dimension they span, making them a cornerstone of vector algebra applied to multidimensional analysis. Knowing whether vectors are perpendicular provides insight into systems of motion, forces, and vector fields.

Vectors

Vectors are foundational elements of mathematics and physics, representing quantities that possess both magnitude and direction. Unlike scalar quantities, vectors add a directional component to equations, representing quantities like force, velocity, and displacement. Here are some key characteristics and operations involving vectors:

• A vector is represented in the form \( \mathbf{v} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} \), where \( x, y, z \) are components along coordinate axes.
• Vectors are often visualized as arrows in 3D space—where the direction of the arrow signifies direction and arrow length conveys magnitude.
• They can be added or subtracted vectorially, respecting both magnitude and direction to yield a resultant vector.
• Understanding vectors helps in situations involving movement or forces acting simultaneously in multiple directions.

Comprehension of vectors and their operations is crucial for anyone pursuing studies in physics, engineering, computer science, or mathematics. They serve as building blocks for complex calculations involving motion, force, and energy in vector space.