Problem 15 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The projections of a vector on the three coordinate axis are \(6,-3,2\) respectively. The direction cosines of the vector are [2009] (a) \(\frac{6}{5}, \frac{-3}{5}, \frac{2}{5}\) (b) \(\frac{6}{7}, \frac{-3}{7}, \frac{2}{7}\) (c) \(\frac{-6}{7}, \frac{-3}{7}, \frac{2}{7}\) (d) \(6,-3,2\)

Step-by-Step Solution

Answer

The direction cosines are option (b): \( \frac{6}{7}, \frac{-3}{7}, \frac{2}{7} \).

Step 1: Understand the Projections

The projections of the vector on the coordinate axes are given as 6, -3, and 2. These are the scalar components of the vector along the x, y, and z axes, respectively. We can represent the vector as \( \mathbf{v} = 6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \) with components \( a = 6 \), \( b = -3 \), and \( c = 2 \).

Step 2: Calculate the Magnitude of the Vector

To find the magnitude of the vector \( \mathbf{v} \), we use the formula for the magnitude of a vector: \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \). Substitute the values to get \( |\mathbf{v}| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7 \).

Step 3: Find the Direction Cosines

The direction cosines of a vector are given by \( \cos \alpha = \frac{a}{|\mathbf{v}|} \), \( \cos \beta = \frac{b}{|\mathbf{v}|} \), and \( \cos \gamma = \frac{c}{|\mathbf{v}|} \). Substituting the values, we get \( \cos \alpha = \frac{6}{7} \), \( \cos \beta = \frac{-3}{7} \), and \( \cos \gamma = \frac{2}{7} \).

Step 4: Match with the Given Options

The calculated direction cosines are \( \frac{6}{7} \), \( \frac{-3}{7} \), \( \frac{2}{7} \). This matches option (b).

Key Concepts

Vector Magnitude

The magnitude of a vector is a measure of how long the vector is. It doesn't matter in which direction the vector points. To find the magnitude of a vector, you can use the Pythagorean formula. This formula is an extension of the Pythagorean Theorem that you might have learned in geometry.For a vector represented as \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \), the magnitude \( |\mathbf{v}| \) is calculated as:

• \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \)

So, when you have the components of a vector (like you do in many problems related to vector directions and magnitudes), you just square each component, add them all up, and take the square root of that sum.Applying this method to the example given where \( a = 6 \), \( b = -3 \), and \( c = 2 \), you get:

• \( |\mathbf{v}| = \sqrt{6^2 + (-3)^2 + 2^2} \)
• \( = \sqrt{36 + 9 + 4} \)
• \( = \sqrt{49} = 7 \)

This shows that the vector is 7 units long. Remember, in physics and engineering, understanding the magnitude helps determine the scale or intensity of the vector.

Projection of Vectors

Projection of vectors is an important concept in understanding how a vector interacts with different dimensions or directions. The projection of a vector onto an axis gives you a sense of the vector's shadow or footprint in that specific direction.Let's say you have a vector \( \mathbf{v} \) and you want to see how it projects onto each of the three coordinate axes. For the example provided, the projections are given as 6, -3, and 2.

• The projection onto the x-axis is 6. This means if you were to "drop a shadow" of the vector in the x-direction, it would extend 6 units.
• The y-axis projection of -3 indicates that in the y-direction, the vector extends negatively or backwards for 3 units.
• For the z-axis, the projection is 2, so the vector points 2 units upwards or downwards, depending on your coordinate system.

When you talk about these projections in vector notation, you write the vector as \( 6\mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \). This makes it easy to visualize how the vector is broken down into its fundamental directions or components.

Coordinate Axes

Coordinate axes are the lines that define the orientation and space of your measurements in a coordinate system. In 3D space, which is common for vectors, you typically deal with three axes:

• The x-axis (left and right)
• The y-axis (up and down)
• The z-axis (gives the third dimension of depth)

These axes form the framework that lets you describe any point or vector in three-dimensional space using coordinates (x, y, z).To better understand vectors within this framework, visualize these axes as the skeletal structure onto which vectors are fixed. The vector's projections onto these axes tell you how the vector stretches along each direction individually. For instance, in the exercise, the vector's projections were \( 6\), \(-3\), and \(2\) for the x, y, and z axes respectively.Understanding the coordinate axes is crucial when dealing with vectors, because every change in direction affects how the vector is represented. Knowing this helps you better grasp how vectors interact with different dimensions and the overall direction of force or motion in physics.