What is the angle between the vector (3i + 3j) and the positive direction of the x-axis?

To determine the angle between the vector 3i + 3j and the positive direction of the x-axis, we can use the concept of the dot product and the formula that relates the two vectors.

First, let’s express the vector similarly to its components. The vector 3i + 3j can be represented as:

  • V = (3, 3)

Now, the positive direction of the x-axis can be represented as the unit vector:

  • U = (1, 0)

To find the angle θ between the two vectors, we can use the following formula:

cos(θ) = (V • U) / (|V| * |U|)

Where:

  • V • U is the dot product of the vectors.
  • |V| is the magnitude of vector V.
  • |U| is the magnitude of vector U.

Let’s compute these values:

  • The dot product: V • U = 3 * 1 + 3 * 0 = 3.
  • Magnitude of vector V: |V| = √(3² + 3²) = √(9 + 9) = √18 = 3√2.
  • Magnitude of vector U: |U| = √(1² + 0²) = √1 = 1.

Now, substitute these values into the cosine formula:

cos(θ) = 3 / (3√2 * 1) = 1 / √2

To find the angle θ, we compute the arccosine:

θ = arccos(1 / √2)

This gives us:

  • θ = 45°

Therefore, the angle between the vector 3i + 3j and the positive direction of the x-axis is 45 degrees.

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