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.
- فلسطين: الصف ١١
- أوكرانيا: الصف ٩
- روسيا: الصف ٩
- ألمانيا: الصف ١٠
- فرنسا: الصف ١٠
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الدليل
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 = ٠°. The angle α can be acute, right or obtuse: ٠° ≤ α < ١٨٠°.
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: k١ = k٢. Lines are perpendicular when k١k٢ = −١.
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 > ٠ they point the same way, if k < ٠ opposite ways, and the length is |k| times the length of a.