Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉

검증된 단계별 안내

Identify the given vectors: \( \mathbf{u} = \langle -1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 0 \rangle \).

Understand that the expression \( \mathbf{v} - 3\mathbf{u} \) means you need to multiply vector \( \mathbf{u} \) by the scalar 3, then subtract the resulting vector from \( \mathbf{v} \).

Calculate the scalar multiplication: multiply each component of \( \mathbf{u} \) by 3, which gives \( 3 \mathbf{u} = \langle 3 \times (-1), 3 \times 2 \rangle = \langle -3, 6 \rangle \).

Perform the vector subtraction by subtracting the corresponding components of \( 3\mathbf{u} \) from \( \mathbf{v} \): \( \mathbf{v} - 3\mathbf{u} = \langle 3 - (-3), 0 - 6 \rangle \).

Simplify the subtraction inside the components to get the resulting vector.

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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 coordinate form, such as u = 〈x, y〉. Understanding how to interpret these components is essential for performing operations like addition, subtraction, and scalar multiplication.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a real number (scalar). For example, multiplying vector u = 〈x, y〉 by scalar 3 results in 〈3x, 3y〉. This operation changes the vector's magnitude but not its direction unless the scalar is negative.

Vector Addition and Subtraction

Adding or subtracting vectors is done component-wise: for vectors a = 〈a1, a2〉 and b = 〈b1, b2〉, a ± b = 〈a1 ± b1, a2 ± b2〉. This principle allows combining vectors or finding the difference between them, which is crucial for solving expressions like v - 3u.

Given vectors u and v, find: v - 3u. u = 〈-1, 2〉, v = 〈3, 0〉