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Vectors
Dynamic tasks on vector coordinates, sum, difference, scalar multiplication, magnitude and equality.
Questions: 10Estimated time: 14 minutes
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Question 1out of 10
Find vector AB when A(1, 7) and B(7, 12).
Vector AB equals endpoint coordinates minus starting coordinates: (xB − xA, yB − yA). Order matters: BA is the opposite vector.
Question 2out of 10
Given points A(-1, -6), B(-2, -6) and C(0, 2), what must the coordinates of D be for vectors AB and CD to be equal?
Find vector AB by subtracting each coordinate of A from the corresponding coordinate of B. Apply the same displacement to C: add the x and y changes separately, keeping their signs. Check that subtracting C from your chosen D gives the same vector.
Question 3out of 10
Find the vector opposite to (8, 4).
The opposite vector has the same magnitude and reversed direction. Both coordinates change sign: (a, b) → (−a, −b).
Question 4out of 10
How do you find the coordinates of a point translated by a given vector?
Add the translation vector to the corresponding point coordinates: (x, y) + (dx, dy) = (x + dx, y + dy).
Question 5out of 10
Add vectors (0, 4) and (6, 3).
Add vectors coordinate by coordinate: (a, b) + (c, d) = (a + c, b + d). Add first coordinates together and second coordinates together.
Question 6out of 10
Multiply vector (0, -2) by 4.
Multiply every coordinate by k: k(a, b) = (ka, kb). A negative k also reverses the direction.
Question 7out of 10
Subtract vector (-5, -6) from (2, 1).
Subtract vectors coordinate by coordinate: (u₁, u₂) − (v₁, v₂) = (u₁ − v₁, u₂ − v₂).
- Subtract the first coordinate of the second vector from the first coordinate of the first vector.
- Do the same with the second coordinates.
- Write the two results as a new ordered pair.
Take particular care with signs: subtracting a negative coordinate becomes addition.
Question 8out of 10
Calculate the magnitude of vector (16, 30).
The magnitude of (a, b) is √(a² + b²), by Pythagoras. Magnitude is always nonnegative.
Question 9out of 10
Multiply vector (-5, -3) by -1.
Multiply every coordinate by k: k(a, b) = (ka, kb). A negative k also reverses the direction.
Question 10out of 10
Add vectors (1, 4) and (1, 6).
Add vectors coordinate by coordinate: (a, b) + (c, d) = (a + c, b + d). Add first coordinates together and second coordinates together.
Time left: 00:00:00
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Sample questions
- Find vector AB when A(___, ___) and B(___, ___).
- Add vectors (___, ___) and (___, ___).
- Subtract vector (___, ___) from (___, ___).
- Multiply vector (___, ___) by ___.
- Find the vector opposite to (___, ___).
Vectors
A vector has direction and magnitude. If A(x₁, y₁) and B(x₂, y₂), then vector AB is (x₂ − x₁, y₂ − y₁). Two vectors are equal when both corresponding coordinates match, regardless of their position in the plane.
Vector operations
Add and subtract vectors coordinate by coordinate. Scalar multiplication multiplies both coordinates, and the magnitude is |v| = √(x² + y²). Pay attention to point order: vector BA is the opposite of vector AB.
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