The gradient of a line and collinear vectors
Find the gradient k from the angle of inclination of a line (k = tan α, including an obtuse α), perpendicular lines and a vector collinear with a given one.
- France: grade 10
- Germany: grade 10
- Palestine: grade 11
- Ukraine: grade 9
- Russia: grade 9
Guide
In Ukrainian Grade 9 geometry (§11) the angle of inclination of the line y = kx + b is the angle α between the line and the positive direction of the x-axis. For a line parallel to the x-axis α = 0°. The angle α can be acute, right or obtuse: 0° ≤ α < 180°.
The gradient of a line equals the tangent of its angle of inclination: k = tanα. A rising line has an acute angle and positive k, a falling line has an obtuse angle and negative k.
Parallel lines have equal gradients: k1 = k2. Lines are perpendicular when k1k2 = −1.
In §15 (multiplying a vector by a number): non-zero vectors a and b are collinear exactly when there is a number k with b = ka. If k > 0 they point the same way, if k < 0 opposite ways, and the length is |k| times the length of a.