INTRODUCTION TO THREE DIMENSIONAL GEOMETRY — MATHS STD 11 Science — Question
Gujarat BoardEnglish MediumSTD 11 ScienceMATHSINTRODUCTION TO THREE DIMENSIONAL GEOMETRY4 Marks
Question
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
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Answer
Let points (–2, 3, 5), (1, 2, 3), and (7, 0, –1) be denoted by P, Q, and R respectively. Points P, Q, and R are collinear if they lie on a line. $\text{PQ}=\sqrt{(1+2)^2+(2-3)^2+(3-5)^2}$ $=\sqrt{(3)^3+(-1)^2+(-2)^2}$ $=\sqrt{9+1+4}$ $=\sqrt{14}$ $\text{QR}=\sqrt{(7-1)^2+(0-2)^2+(-1-3)^2}$ $=\sqrt{(6)^2+(-2)^2+(-4)^2}$ $=\sqrt{36+4+16}$ $=\sqrt{56}$ $=2\sqrt{14}$ $\text{PR}=\sqrt{(7+2)^2+(0+3)^2+(-1-5)^2}$ $=\sqrt{(9)^2+(-3)^2+(-6)^2}$ $=\sqrt{81+9+36}$ $=\sqrt{126}$ $=3\sqrt{14}$ Here, $\text{PQ + QR}=\sqrt{14}+2\sqrt{14}=3\sqrt{14}=\text{PR}$ Hence, points P(–2, 3, 5), Q(1, 2, 3), and R(7, 0, –1) are collinear.
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