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

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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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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.

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Position Vectors & Component Form

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.

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Introduction to Dot Product

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.

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