In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = i - j

검증된 단계별 안내

Identify the given vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), which can be written in component form as \( \mathbf{v} = \langle 1, -1 \rangle \).

Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula \( \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2} \). Substitute the components to get \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).

Simplify the expression under the square root to find the magnitude \( \| \mathbf{v} \| \).

Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{v} = \left\langle \frac{1}{\| \mathbf{v} \|}, \frac{-1}{\| \mathbf{v} \|} \right\rangle \).

Express the unit vector \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{i} - \frac{1}{\| \mathbf{v} \|} \mathbf{j} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vector Direction

The direction of a vector is the orientation it points to in space, independent of its length. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction is essential to find a unit vector that aligns with the original vector.

Vector Magnitude (Length)

The magnitude of a vector is its length, calculated using the square root of the sum of the squares of its components. For vector v = i - j, the magnitude is √(1² + (-1)²) = √2. This value is used to normalize the vector.

Unit Vector

A unit vector has a magnitude of 1 and points in a specific direction. To find a unit vector in the same direction as v, divide each component of v by its magnitude. This process scales the vector to length one without changing its direction.

In Exercises 39–46, find the unit vector that has the same direction as the vector v.v = i - j