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QLDSpecialist MathematicsQuick questions

Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

Quick questions on Vector and Cartesian equations of lines in QCE Specialist Mathematics Unit 3

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is vector equation of a line?
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A line through a point with position vector a\mathbf{a} and direction d\mathbf{d} has vector equation
What is cartesian equation?
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Eliminating the parameter tt (when each di≠0d_i \neq 0) gives the symmetric Cartesian form
What is solve the first two?
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From the second equation, s=1−ts = 1 - t. Substitute into the first: 1+2t=3+(1−t)1 + 2t = 3 + (1 - t), so 1+2t=4−t1 + 2t = 4 - t, giving 3t=33t = 3 and t=1t = 1. Then s=1−1=0s = 1 - 1 = 0.
What is test the third equation?
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With t=1t = 1 and s=0s = 0: LHS =2−1=1= 2 - 1 = 1, RHS =2(0)=0= 2(0) = 0. Since 1≠01 \neq 0, the third equation fails.
What is conclusion?
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The directions (2,1,−1)(2,1,-1) and (1,−1,2)(1,-1,2) are not scalar multiples, so the lines are not parallel, and the system is inconsistent, so they do not intersect. The lines are skew.

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