Find a vector that points in the same direction as the vector ( î ＋ ĵ ) and whose magnitude is 1.

Verified step by step guidance

1. To find a unit vector (a vector with magnitude 1) in the same direction as a given vector, we first calculate the magnitude of the given vector. The magnitude of a vector \( \mathbf{v} = a\hat{i} + b\hat{j} \) is given by \( |\mathbf{v}| = \sqrt{a^2 + b^2} \). For the vector \( \mathbf{v} = \hat{i} + \hat{j} \), substitute \( a = 1 \) and \( b = 1 \) into the formula.
2. Simplify the magnitude formula: \( |\mathbf{v}| = \sqrt{1^2 + 1^2} = \sqrt{2} \). This is the magnitude of the vector \( \hat{i} + \hat{j} \).
3. To create a unit vector, divide each component of the original vector by its magnitude. The formula for the unit vector \( \mathbf{u} \) is \( \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \). Substituting \( \mathbf{v} = \hat{i} + \hat{j} \) and \( |\mathbf{v}| = \sqrt{2} \), we get \( \mathbf{u} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \).
4. Simplify the components of the unit vector: \( \mathbf{u} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \). This is the unit vector in the same direction as \( \hat{i} + \hat{j} \).
5. Verify that the magnitude of the unit vector is indeed 1 by calculating \( |\mathbf{u}| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \). This confirms that the vector \( \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \) is a unit vector.

Key Concepts

Vector Direction

The direction of a vector is defined by the angle it makes with a reference axis, typically represented in a Cartesian coordinate system. In this case, the vector (î + ĵ) points diagonally in the first quadrant, indicating equal contributions from both the x and y components. Understanding vector direction is crucial for determining how to scale a vector while maintaining its orientation.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector represented as (a, b), the magnitude is given by √(a² + b²). In the context of the vector (î + ĵ), its magnitude is √(1² + 1²) = √2. This concept is essential for normalizing a vector to a unit vector with a magnitude of 1.

Unit Vector

A unit vector is a vector that has a magnitude of exactly 1 and indicates direction only. To convert any vector into a unit vector, you divide each component of the vector by its magnitude. For the vector (î + ĵ), the unit vector can be found by dividing each component by √2, resulting in (1/√2, 1/√2). This process is fundamental in various applications, including physics and engineering, where direction is important but magnitude needs to be standardized.