Let A = 4i - 2j, B = -3i + 5j, and D = A - B. Draw a coordinate system and on it show vectors A, B, and D.

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Start by understanding the given vectors: A = 4i - 2j and B = -3i + 5j. These are expressed in unit vector notation, where 'i' represents the x-direction and 'j' represents the y-direction.

To find vector D, use the vector subtraction formula: D = A - B. Subtract the corresponding components of vectors A and B. For the x-components: 4 - (-3). For the y-components: -2 - 5.

Simplify the subtraction to determine the components of vector D. The x-component becomes 4 + 3, and the y-component becomes -2 - 5. This gives the final form of vector D in terms of its components.

Once you have the components of D, represent it in unit vector notation as D = (x-component)i + (y-component)j. This will give you the complete expression for vector D.

To draw the coordinate system, plot vectors A, B, and D on a Cartesian plane. Start each vector at the origin, and use their components to determine their endpoints. For example, vector A starts at (0, 0) and ends at (4, -2). Similarly, plot B and D using their respective components.

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주요 개념

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Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system using components along the axes. In this case, vectors A and B are expressed in terms of their i (horizontal) and j (vertical) components, allowing for easy visualization and manipulation in a two-dimensional space.

Vector Addition

Vector addition involves combining two or more vectors to find a resultant vector. This is done by adding their corresponding components. For example, to find vector D, we subtract vector B from vector A, which involves subtracting the i and j components separately to determine the new vector's components.

Coordinate System

A coordinate system provides a framework for locating points and vectors in space. In two dimensions, the Cartesian coordinate system uses perpendicular axes (x and y) to define positions. When drawing vectors A, B, and D, the coordinate system helps visualize their directions and magnitudes relative to each other.