Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u

검증된 단계별 안내

Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).

Recall that vector subtraction \( \mathbf{v} - \mathbf{u} \) is performed component-wise: subtract the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \).

Set up the subtraction for each component: \( (v_x - u_x, v_y - u_y) \), where \( v_x = 4, u_x = -2, v_y = 3, u_y = 5 \).

Calculate each component difference separately: \( 4 - (-2) \) for the x-component and \( 3 - 5 \) for the y-component.

Combine the results to write the vector \( \mathbf{v} - \mathbf{u} \) as \( \langle 4 - (-2), 3 - 5 \rangle \).

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주요 개념

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Vector Representation

Vectors are quantities defined by both magnitude and direction, often represented as ordered pairs or tuples in two dimensions. For example, u = 〈-2, 5〉 indicates a vector with components -2 along the x-axis and 5 along the y-axis.

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. For vectors v = 〈v₁, v₂〉 and u = 〈u₁, u₂〉, the difference v - u is 〈v₁ - u₁, v₂ - u₂〉, resulting in a new vector.

Component-wise Operations

Operations on vectors such as addition and subtraction are performed component-wise, meaning each component is handled independently. This simplifies calculations and helps visualize vector operations geometrically.

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.v - u