J: Vectors: Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.

The idea

A vector in component form is written as a column pair (x, y) or as xi + yj, where x and y are the horizontal and vertical components. Its magnitude is the length of the arrow, found using Pythagoras: |v| = sqrt(x^2 + y^2). Its direction is the angle measured anticlockwise from the positive x-axis, usually in degrees. Find the acute angle alpha with the x-axis using tan(alpha) = |y|/|x|, then adjust for the quadrant: in the second quadrant use 180 - alpha, in the third use 180 + alpha, and in the fourth use 360 - alpha. For example, (-1, 1) has alpha = 45 degrees, so its direction is 180 - 45 = 135 degrees. Converting back, x = r cos(theta) and y = r sin(theta), where r is the magnitude and theta the direction. For example, a vector of magnitude 10 at 210 degrees has components (-8.66, -5). Always check the quadrant and give direction in the range 0 to 360 degrees unless told otherwise.

What earns the marks

• Use Pythagoras correctly: magnitude r = sqrt(x^2 + y^2), showing the squares and the square root.
• Find the reference angle with tan(alpha) = |y|/|x| and then place it in the correct quadrant using 180 - alpha, 180 + alpha or 360 - alpha.
• Convert from magnitude/direction to components using x = r cos(theta) and y = r sin(theta).
• State direction in the required range, typically 0 to 360 degrees, with degrees or units as appropriate.
• Check the answer by sketching the vector or substituting back into the component equations.

Common mistakes and how to put them right

• Forgetting to square both components before adding.

  Put it right: Apply Pythagoras to the two perpendicular components.

• Using tan(theta) = y/x and accepting the calculator's angle without quadrant adjustment.

  Put it right: Identify the quadrant from the signs of x and y, then use 180 - alpha, 180 + alpha or 360 - alpha as appropriate.

• Mixing up sine and cosine when converting back.

  Put it right: x is adjacent to theta, so x = r cos(theta), and y is opposite, so y = r sin(theta).

Exam tips

• Sketch the vector on axes before calculating; the sketch prevents quadrant errors and helps you check the size of the angle.
• If a question asks for direction as a bearing, measure clockwise from north instead of anticlockwise from the positive x-axis.