Find the dot product for each pair of vectors.
4i, 5i - 9j

검증된 단계별 안내

Identify the given vectors. The first vector is \(\vec{A} = 4\mathbf{i}\) and the second vector is \(\vec{B} = 5\mathbf{i} - 9\mathbf{j}\).

Recall the formula for the dot product of two vectors \(\vec{A} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\vec{B} = b_1\mathbf{i} + b_2\mathbf{j}\): \(\vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2\).

Extract the components of each vector: For \(\vec{A}\), \(a_1 = 4\) and \(a_2 = 0\) (since there is no \(\mathbf{j}\) component). For \(\vec{B}\), \(b_1 = 5\) and \(b_2 = -9\).

Substitute the components into the dot product formula: \(\vec{A} \cdot \vec{B} = (4)(5) + (0)(-9)\).

Simplify the expression by multiplying and adding the terms to find the dot product.

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주요 개념

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

Vector Components and Notation

Vectors are often expressed in terms of unit vectors i, j, and k, representing the x, y, and z directions respectively. Understanding how to identify and separate these components is essential for performing operations like the dot product.

Dot Product Definition

The dot product of two vectors is a scalar obtained by multiplying corresponding components and summing the results. For vectors in two dimensions, it is calculated as (x1 * x2) + (y1 * y2), which measures the extent to which the vectors point in the same direction.

Properties and Applications of the Dot Product

The dot product is commutative and relates to the angle between vectors through the formula A · B = |A||B|cosθ. It is used to find projections, angles, and to determine orthogonality between vectors.

Find the dot product for each pair of vectors.4i, 5i - 9j