In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j. P₁ = (-3, 4), P₂ = (6, 4)

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Identify the coordinates of the initial point \( P_1 = (-3, 4) \) and the terminal point \( P_2 = (6, 4) \).

Calculate the change in the x-coordinate: \( \Delta x = x_2 - x_1 = 6 - (-3) \).

Calculate the change in the y-coordinate: \( \Delta y = y_2 - y_1 = 4 - 4 \).

Express the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the changes in coordinates: \( \mathbf{v} = \Delta x \mathbf{i} + \Delta y \mathbf{j} \).

Substitute the calculated values of \( \Delta x \) and \( \Delta y \) into the expression for \( \mathbf{v} \).

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주요 개념

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Vectors

A vector is a mathematical object that has both magnitude and direction. In a two-dimensional space, a vector can be represented as an ordered pair (x, y), where x and y denote the horizontal and vertical components, respectively. Understanding vectors is essential for solving problems involving direction and displacement between points.

Unit Vectors i and j

In the context of two-dimensional vectors, 'i' and 'j' are the standard unit vectors that represent the x-axis and y-axis directions, respectively. The vector 'i' corresponds to (1, 0) and 'j' corresponds to (0, 1). Any vector can be expressed as a linear combination of these unit vectors, which simplifies vector representation and calculations.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be interpreted as determining the displacement from one point to another. For points P₁ and P₂, the vector v can be calculated by subtracting the coordinates of P₁ from those of P₂. This operation yields a new vector that indicates the direction and distance from P₁ to P₂.