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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 6 Find the vector and the Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector 3𝑖 ̂ + 2𝑗 ̂ – 8𝑘 ̂ . Vector equation Equation of a line passing through a point with position vector 𝑎 ⃗ , and parallel to a vector 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆𝒃 ⃗ Since line passes through (5, 2, −4) 𝒂 ⃗ = 5𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂ Since line is parallel to 3𝒊 ̂ + 2𝒋 ̂ − 8𝒌 ̂ 𝒃 ⃗ = 3𝑖 ̂ + 2𝑗 ̂ − 8𝑘 ̂ Equation of line 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ 𝒓 ⃗ = (5𝒊 ̂ + 2𝒋 ̂ − 4𝒌 ̂) + 𝜆 (3𝒊 ̂ + 2𝒋 ̂ − 8𝒌 ̂) Cartesian equation Equation of a line passing through a point (x, y, z) and parallel to a line with direction ratios a, b, c is (𝒙 − 𝒙𝟏)/𝒂 = (𝒚 − 𝒚𝟏)/𝒃 = (𝒛 − 𝒛𝟏)/𝒄 Since line passes through (5, 2, −4) 𝒙1 = 5, y1 = 2 , z1 = −4 Also, line is parallel to 3𝒊 ̂ + 2𝒋 ̂ −8𝒌 ̂ , 𝒂 = 3, b = 2, c = −8 Equation of line in Cartesian form is (𝑥 − 𝑥1)/𝑎 = (𝑦 − 𝑦1)/𝑏 = (𝑧 − 𝑧1)/𝑐 (𝑥 − 5)/3 = (𝑦 − 2)/2 = (𝑧 − (−4))/( −8) (𝒙 − 𝟓)/𝟑 = (𝒚 − 𝟐)/𝟐 = (𝒛 + 𝟒)/(−𝟖)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.