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Last updated at May 29, 2023 by Teachoo
Question 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 So, 𝐴_1/𝐴_2 = 2/(−1) = −2, 𝐵_1/𝐵_2 = 1/2 , 𝐶_1/𝐶_2 = 3/0 Since, direction ratios are not proportional, the two normal are not parallel. ∴ Given two planes are not parallel. Check perpendicular Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are perpendicular if 𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐 = 0 Now, 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = (2 × −1) + (1 × 2) + (3 × 0) = −2 + 2 + 0 = 0 Since 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = 0 The two normal are perpendicular. Since normal are perpendicular, planes are perpendicular.