In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j

검증된 단계별 안내

Identify the components of the vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \). Here, the vector has an \( x \)-component of 1 (from \( \mathbf{i} \)) and a \( y \)-component of -1 (from \( -\mathbf{j} \)). So, \( \mathbf{v} = (1, -1) \).

To sketch the vector as a position vector, start at the origin \( (0,0) \) on the coordinate plane.

From the origin, move 1 unit in the positive \( x \)-direction (right) and 1 unit in the negative \( y \)-direction (down). Mark the point \( (1, -1) \).

Draw an arrow from the origin \( (0,0) \) to the point \( (1, -1) \). This arrow represents the vector \( \mathbf{v} \).

To find the magnitude of \( \mathbf{v} \), use the formula for the length of a vector: \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \). Substitute \( x = 1 \) and \( y = -1 \) to get \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Position Vector

A position vector represents the location of a point in space relative to the origin. It is expressed in terms of unit vectors along coordinate axes, such as i and j in two dimensions, indicating horizontal and vertical components respectively.

Vector Components

Vector components break down a vector into its horizontal (i) and vertical (j) parts. For v = i - j, the vector moves one unit right and one unit down, which helps in visualizing and sketching the vector accurately on the coordinate plane.

Magnitude of a Vector

The magnitude of a vector is its length, calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. For v = i - j, magnitude = √(1² + (-1)²) = √2, representing the distance from the origin to the vector's endpoint.