Introduction to Three-Dimensional Geometry — Practice

Question 1

Find the distance between the points A(1, 2, 3) and B(4, 6, 3).

Accepted Answer

AB = √[(4−1)²+(6−2)²+(3−3)²] = √[9+16+0] = √25 = 5.

Question 2

Find the coordinates of the point that divides the line segment joining A(1, −2, 3) and B(3, 4, −5) internally in the ratio 2:3.

Accepted Answer

Using section formula with m=2, n=3: x = (2×3+3×1)/(2+3) = (6+3)/5 = 9/5. y = (2×4+3×(−2))/5 = (8−6)/5 = 2/5. z = (2×(−5)+3×3)/5 = (−10+9)/5 = −1/5. Point = (9/5, 2/5, −1/5).

Question 3

Show that the points A(1, 2, 3), B(−1, −2, −1), C(2, 3, 2) are vertices of a triangle and find which type by computing all three side lengths.

Accepted Answer

AB = √[(−1−1)²+(−2−2)²+(−1−3)²] = √[4+16+16] = √36 = 6. BC = √[(2+1)²+(3+2)²+(2+1)²] = √[9+25+9] = √43. AC = √[(2−1)²+(3−2)²+(2−3)²] = √[1+1+1] = √3. Since AB ≠ BC ≠ AC and AB² ≠ BC²+AC² (6²=36 ≠ 43+3=46), the triangle is scalene and not right-angled. All sides are different, so it is a scalene triangle.

Practice — Introduction to Three-Dimensional Geometry

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