Check sibling questions

Ex 11.3, 13 (b) - Find angle b/w planes 2x+y+3z-2=0 and x - 2y + 5 = 0

Ex 11.3, 13 (b) - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Ex 11.3, 13 (b) - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Transcript

Ex 11.3, 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, 𝐡_1, 𝐢_1 and 𝐴_2, 𝐡_2, 𝐢_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = π‘ͺ_𝟏/π‘ͺ_𝟐 So, 𝐴_1/𝐴_2 = 2/(βˆ’1) = βˆ’2, 𝐡_1/𝐡_2 = 1/2 , 𝐢_1/𝐢_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, 𝐡_1, 𝐢_1 and 𝐴_2, 𝐡_2, 𝐢_2 are perpendicular if 𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + π‘ͺ_𝟏 π‘ͺ_𝟐 = 0 Now, 𝐴_1 𝐴_2 + 𝐡_1 𝐡_2 + 𝐢_1 𝐢_2 = (2 Γ— βˆ’1) + (1 Γ— 2) + (3 Γ— 0) = βˆ’2 + 2 + 0 = 0 Since 𝐴_1 𝐴_2 + 𝐡_1 𝐡_2 + 𝐢_1 𝐢_2 = 0 The two normal are perpendicular. Since normal are perpendicular, planes are perpendicular.

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