In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

검증된 단계별 안내

Identify the coordinates of the points given in the vector. Here, the vector \( \mathbf{b} \) starts at \((-21, 10)\) and ends at \((-21, -20)\).

Calculate the components of the vector \( \mathbf{b} \) by subtracting the coordinates of the initial point from the terminal point: \( \mathbf{b} = (-21 - (-21), -20 - 10) = (0, -30) \).

Since vectors \( \mathbf{u} \) and \( \mathbf{v} \) have the same direction, the magnitude of \( \mathbf{v} \) can be found by calculating the length of \( \mathbf{b} \), which is the magnitude of the vector between the two points.

Use the distance formula (or magnitude formula for vectors) to find the length of \( \mathbf{b} \): \[ ||\mathbf{b}|| = \sqrt{(0)^2 + (-30)^2} \].

Simplify the expression to find the magnitude \( ||\mathbf{b}|| \), which is the same as \( ||\mathbf{v}|| \) since they have the same direction.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Vector Magnitude (Norm)

The magnitude or norm of a vector is the length of the vector in the coordinate plane. It is calculated using the distance formula derived from the Pythagorean theorem, which for a vector with components (x, y) is ||v|| = √(x² + y²). This represents the straight-line distance from the origin to the point (x, y).

Vector Direction and Components

A vector's direction is determined by the angle it makes with the coordinate axes, and its components represent its projection along these axes. When two vectors have the same direction, one is a scalar multiple of the other, meaning their components are proportional. Understanding this helps in relating vectors u and v in the problem.

Using Coordinates to Find Vector Components

Vectors can be represented by points in the plane, where the vector from point A to point B is found by subtracting coordinates: (x2 - x1, y2 - y1). This allows calculation of the vector's components, which are essential for finding its magnitude and direction.