A vector \(\vec r=a \hat i+b \hat j\) is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?
√2, √2
Understanding the Problem: Vector Equally Inclined to Axes
The question asks us to find the values of the components 'a' and 'b' of a 2D vector \(\vec r = a \hat i + b \hat j\). We are given two key pieces of information about this vector:
- The vector is equally inclined to both the x and y axes.
- The magnitude of the vector is 2 units.
Let's break down these conditions to solve for 'a' and 'b'.
Analyzing Equally Inclined Vectors in 2D
A vector \(\vec r = a \hat i + b \hat j\) makes an angle \(\alpha\) with the positive x-axis and an angle \(\beta\) with the positive y-axis. The direction cosines are given by:
- \(\cos \alpha = \dfrac{a}{|\vec r|}\)
- \(\cos \beta = \dfrac{b}{|\vec r|}\)
The condition that the vector is equally inclined to both x and y axes means that the angles \(\alpha\) and \(\beta\) have the same magnitude. In the context of the axes, this usually implies that the direction cosines are equal in magnitude, i.e., \(|\cos \alpha| = |\cos \beta|\), which simplifies to \(|a| = |b|\).
For a vector in the first quadrant (where 'a' and 'b' are positive), being equally inclined means the angle with both positive axes is 45 degrees (\(45^\circ\)). In this case, \(a = b\). Looking at the options provided, both 'a' and 'b' are positive, so we can assume \(a=b\).
Using the Magnitude of the Vector
The magnitude of a 2D vector \(\vec r = a \hat i + b \hat j\) is given by the formula:
\(|\vec r| = \sqrt{a^2 + b^2}\)
We are given that the magnitude of the vector is 2 units. So, we have the equation:
\(\sqrt{a^2 + b^2} = 2\)
Solving for a and b
We have two conditions:
- \(a = b\) (from the equally inclined condition, assuming positive components)
- \(\sqrt{a^2 + b^2} = 2\) (from the magnitude condition)
Substitute the first condition (\(a=b\)) into the second equation:
\(\sqrt{a^2 + a^2} = 2\)
\(\sqrt{2a^2} = 2\)
Now, let's solve for 'a'. We can square both sides of the equation:
\((\sqrt{2a^2})^2 = 2^2\)
\(2a^2 = 4\)
Divide by 2:
\(a^2 = \dfrac{4}{2}\)
\(a^2 = 2\)
Taking the square root of both sides:
\(a = \pm \sqrt{2}\)
Since we assumed positive components based on the options, we take the positive value:
\(a = \sqrt{2}\)
Since \(a = b\), we also have:
\(b = \sqrt{2}\)
Thus, the values of a and b are \(\sqrt{2}\) and \(\sqrt{2}\) respectively.
Verification
Let's check if the vector \(\vec r = \sqrt{2} \hat i + \sqrt{2} \hat j\) satisfies the given conditions:
- Magnitude: \(|\vec r| = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2\). The magnitude is indeed 2 units.
- Equally Inclined: The components are \(a = \sqrt{2}\) and \(b = \sqrt{2}\). Since \(a=b\), the vector makes an angle of \(45^\circ\) with the positive x-axis and \(45^\circ\) with the positive y-axis, meaning it is equally inclined.
The values \(a = \sqrt{2}\) and \(b = \sqrt{2}\) satisfy both conditions.
Final Answer
The values of a and b are \(\sqrt{2}\) and \(\sqrt{2}\).
Revision Table: Key Concepts for Vector Problems
| Concept | Description | Formula/Property |
|---|---|---|
| Vector Components | Representation of a vector along coordinate axes. For \(\vec r = a \hat i + b \hat j\), 'a' is the x-component, 'b' is the y-component. | \(\vec r = a \hat i + b \hat j\) |
| Magnitude of a 2D Vector | The length or size of the vector. | \(|\vec r| = \sqrt{a^2 + b^2}\) |
| Direction Cosines (in 2D) | Cosines of the angles the vector makes with the positive coordinate axes. | \(\cos \alpha = \dfrac{a}{|\vec r|}, \cos \beta = \dfrac{b}{|\vec r|}\) |
| Equally Inclined Vector (in 2D, positive components) | A vector making the same angle with the positive x-axis and positive y-axis. This angle is \(45^\circ\). | \(a = b\) |
Additional Information: Exploring Equally Inclined Vectors
The condition "equally inclined to both x and y axes" in 2D means the angle with the x-axis and the angle with the y-axis are the same in magnitude. While in the first quadrant this implies \(a=b\), a vector equally inclined can lie in other quadrants as well. For example:
- If \(a = -\sqrt{2}\) and \(b = \sqrt{2}\), the vector \(\vec r = -\sqrt{2} \hat i + \sqrt{2} \hat j\) has magnitude \(\sqrt{(-\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2+2} = 2\). This vector is equally inclined to the negative x-axis and positive y-axis. The components satisfy \(|a| = |b|\).
- If \(a = -\sqrt{2}\) and \(b = -\sqrt{2}\), the vector \(\vec r = -\sqrt{2} \hat i - \sqrt{2} \hat j\) has magnitude \(\sqrt{(-\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2+2} = 2\). This vector is equally inclined to the negative x-axis and negative y-axis. The components satisfy \(|a| = |b|\).
Since the options only provided positive values for 'a' and 'b', we focused on the case where both components are positive, leading to \(a=b=\sqrt{2}\).
Understanding the geometric interpretation of equally inclined vectors helps in visualizing the direction of the vector.
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