Find the angle between each pair of vectors. Round to two decimal places as necessary.
3i + 4j, j

검증된 단계별 안내

Identify the two vectors given: the first vector is \(\mathbf{v_1} = 3\mathbf{i} + 4\mathbf{j}\) and the second vector is \(\mathbf{v_2} = \mathbf{j}\), which can be written as \(0\mathbf{i} + 1\mathbf{j}\).

Recall the formula for the angle \(\theta\) between two vectors \(\mathbf{a}\) and \(\mathbf{b}\): \(\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}\), where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product and \(\|\mathbf{a}\|\), \(\|\mathbf{b}\|\) are the magnitudes of the vectors.

Calculate the dot product of \(\mathbf{v_1}\) and \(\mathbf{v_2}\): \(\mathbf{v_1} \cdot \mathbf{v_2} = (3)(0) + (4)(1) = 4\).

Find the magnitudes of each vector: \(\|\mathbf{v_1}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16}\), \(\|\mathbf{v_2}\| = \sqrt{0^2 + 1^2} = 1\).

Substitute the dot product and magnitudes into the cosine formula and solve for \(\theta\): \(\cos(\theta) = \frac{4}{\|\mathbf{v_1}\| \times 1}\), then find \(\theta\) by taking the inverse cosine (arccos) of the result. Finally, round your answer to two decimal places.

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주요 개념

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

Vector Components and Representation

Vectors in two dimensions can be expressed using unit vectors i and j, representing the x and y components respectively. For example, the vector 3i + 4j has components (3, 4), which helps in calculating magnitude and direction.

Dot Product of Vectors

The dot product of two vectors is a scalar value found by multiplying corresponding components and summing the results. It is used to determine the angle between vectors through the formula: dot product = |A||B|cos(θ).

Calculating the Angle Between Vectors

The angle θ between two vectors can be found using the dot product formula: θ = arccos[(A·B) / (|A||B|)]. This requires computing the dot product and magnitudes of both vectors, then applying the inverse cosine function.

Find the angle between each pair of vectors. Round to two decimal places as necessary.3i + 4j, j