You are given two vectors A = -3i + 6j and B = 7i + 2j. Let Counterclockwise angles be positive. What angle does A make with the +x-axis?

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First, understand that the angle a vector makes with the +x-axis can be found using the tangent function. The tangent of the angle θ is the ratio of the y-component to the x-component of the vector.

For vector A = -3i + 6j, identify the x-component as -3 and the y-component as 6.

Use the formula for the tangent of the angle: tan(θ) = y/x. Substitute the values from vector A into this formula: tan(θ) = 6 / -3.

Calculate the angle θ using the inverse tangent function (arctan). Since the tangent value is negative, the angle will be in the second quadrant. Use θ = arctan(6 / -3).

Remember that angles in the second quadrant are calculated as θ = 180° + arctan(6 / -3) to ensure the angle is counterclockwise from the +x-axis.

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주요 개념

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Vector Components

Vectors are mathematical entities with both magnitude and direction, represented by components along coordinate axes. In this problem, vector A is expressed in terms of its i (x-axis) and j (y-axis) components, which are -3 and 6, respectively. Understanding these components is crucial for calculating the angle with the x-axis.

Dot Product

The dot product of two vectors is a scalar value that can be used to find the angle between them. It is calculated as the sum of the products of their corresponding components. For vector A and the x-axis unit vector, the dot product helps determine the cosine of the angle between them, which is essential for finding the angle itself.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to find angles from known trigonometric ratios. Once the cosine of the angle between vector A and the x-axis is determined using the dot product, the arccosine function can be applied to find the actual angle, considering the sign and direction specified in the problem.