Find the angle between each pair of vectors. Round to two decimal places as necessary.
2i + 2j, -5i - 5j

검증된 단계별 안내

Identify the given vectors: \( \vec{A} = 2\mathbf{i} + 2\mathbf{j} \) and \( \vec{B} = -5\mathbf{i} - 5\mathbf{j} \).

Recall the formula for the angle \( \theta \) between two vectors \( \vec{A} \) and \( \vec{B} \): \[ \cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{\|\vec{A}\| \|\vec{B}\|} \] where \( \vec{A} \cdot \vec{B} \) is the dot product and \( \|\vec{A}\|, \|\vec{B}\| \) are the magnitudes of the vectors.

Calculate the dot product \( \vec{A} \cdot \vec{B} \) using the components: \[ \vec{A} \cdot \vec{B} = (2)(-5) + (2)(-5) \].

Find the magnitudes of each vector: \[ \|\vec{A}\| = \sqrt{2^2 + 2^2} \quad \text{and} \quad \|\vec{B}\| = \sqrt{(-5)^2 + (-5)^2} \].

Substitute the dot product and magnitudes into the cosine formula, then use the inverse cosine function to find the angle \( \theta = \cos^{-1} \left( \frac{\vec{A} \cdot \vec{B}}{\|\vec{A}\| \|\vec{B}\|} \right) \). 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 as combinations of unit vectors i and j, representing the x and y components respectively. Understanding how to interpret and manipulate these components is essential for calculating vector operations such as dot product and magnitude.

Dot Product of Vectors

The dot product is a scalar value obtained by multiplying corresponding components of two vectors and summing the results. It is used to find the angle between vectors through the formula: dot product = |A||B|cos(θ), linking algebraic and geometric interpretations.

Calculating the Angle Between Vectors

The angle θ between two vectors can be found by rearranging 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 to determine the angle in degrees or radians.

Find the angle between each pair of vectors. Round to two decimal places as necessary.2i + 2j, -5i - 5j