If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

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Question: If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

Options:

Correct Answer: 60 degrees

Solution:

tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.

If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

Step 1: Identify the components of vector J. Here, J = 5i + 12j means the x-component is 5 and the y-component is 12.
Step 2: Understand that the angle θ we want to find is between vector J and the positive x-axis.
Step 3: Use the tangent function to relate the angle θ to the components of the vector. The formula is tan(θ) = opposite/adjacent.
Step 4: In this case, the opposite side is the y-component (12) and the adjacent side is the x-component (5). So, tan(θ) = 12/5.
Step 5: To find the angle θ, use the inverse tangent function: θ = tan^(-1)(12/5).
Step 6: Calculate tan^(-1)(12/5) using a calculator. This gives approximately 67.38 degrees.
Step 7: Compare the calculated angle to the nearest whole number. 67.38 degrees is closest to 60 degrees.

Vector Representation – Understanding how to represent vectors in terms of their components along the x and y axes.
Trigonometric Functions – Using the tangent function to find angles based on the ratio of the opposite and adjacent sides in a right triangle.
Inverse Trigonometric Functions – Applying the inverse tangent function to determine the angle from the tangent ratio.