Video Transcript
Find the vector form of the equation of the straight line four 𝑥 minus three over negative nine is equal to seven 𝑦 minus eight over negative two, which is equal to seven plus six 𝑧 over four.
We begin by recalling that the vector form of the equation of a straight line is 𝐫 is equal to 𝐫 sub zero plus 𝑡 multiplied by 𝐝, where 𝐫 sub zero is the position vector of a point that lies on the line. And 𝐝 is the direction vector of the line. In this question, we are given the equation of the line in Cartesian form. We know that the general Cartesian form is 𝑥 minus 𝑥 sub zero over 𝑙 is equal to 𝑦 minus 𝑦 sub zero over 𝑚, which is equal to 𝑧 minus 𝑧 sub zero over 𝑛, where the direction vector is 𝑙, 𝑚, 𝑛 and the position vector of a point that lies on the line is 𝑥 sub zero, 𝑦 sub zero, 𝑧 sub zero.
Our equation is very similar to this general form. However, instead of 𝑥, we have four 𝑥. The 𝑦-term is seven 𝑦 and the 𝑧-term is six 𝑧. If we divide the numerator and denominator of the first expression by four, we have 𝑥 minus three-quarters divided by negative nine over four. Dividing the numerator and denominator of the second expression by seven gives us 𝑦 minus eight over seven divided by negative two over seven. Dividing the numerator and denominator of the third expression by six gives us 𝑧 plus seven over six divided by four over six. We now have an equation that matches the general form of the Cartesian equation.
The third denominator four-sixths simplifies to two-thirds. We now have values of 𝑙, 𝑚, and 𝑛 together with 𝑥 sub zero, 𝑦 sub zero, and 𝑧 sub zero that we can substitute into the vector form. We have a point with coordinates three-quarters, eight-sevenths, and negative seven-sixths that lies on our line. We can substitute these values for 𝐫 sub zero. A direction vector of the line is negative nine-quarters, negative two-sevenths, two-thirds.
The vector form of the equation of the straight line four 𝑥 minus three over negative nine, which is equal to seven 𝑦 minus eight over negative two, which is equal to seven plus six 𝑧 over four, is 𝐫 is equal to three-quarters, eight-sevenths, minus seven-sixths plus 𝑡 multiplied by negative nine-quarters, negative two-sevenths, two-thirds.
