
Q1 If a line makes angles 90°, 135°, 45° with x, y and zaxes respectively, find its direction cosines. Ans: Let direction cosines of the line be l, m, and n.
\begin{align}l = cos90^0=0\end{align}
\begin{align}m = cos135^0=\frac{1}{\sqrt{2}}\end{align}
\begin{align}n = cos45^0=\frac{1}{\sqrt{2}}\end{align}
\begin{align}Therefore, the\; direction\; cosines\; of \;the\; line\; are\;0, \frac{1}{\sqrt{2}}\;and\;\frac{1}{\sqrt{2}}\end{align}
Q2 Find the direction cosines of a line which makes equal angles with the coordinate axes. Ans: Let the direction cosines of the line make an angle α with each of the coordinate axes.
∴ l = cos α, m = cos α, n = cos α
l^{2}+m^{2}+n^{2} =1
⇒ cos^{2}α + cos^{2}α + cos^{2}α = 1
⇒ 3cos^{2}α =1
\begin{align}\Rightarrow cos^2α = \frac{1}{3}\end{align}
\begin{align}\Rightarrow cosα = \pm\frac{1}{\sqrt 3}\end{align}
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are
\begin{align} \pm\frac{1}{\sqrt 3},\pm\frac{1}{\sqrt 3},and \pm\frac{1}{\sqrt 3}\end{align}
Q3 If a line has the direction ratios −18, 12, −4, then what are its direction cosines? Ans: If a line has direction ratios of −18, 12, and −4, then its direction cosines are
\begin{align} \frac{18}{\sqrt {(18)^2 + (12)^2 + (4)^2}},\frac{12}{\sqrt {(18)^2 + (12)^2 + (4)^2}},\frac{4}{\sqrt {(18)^2 + (12)^2 + (4)^2}}\end{align}
\begin{align} i.e., \frac{18}{22},\frac{12}{22},\frac{4}{22}\end{align}
\begin{align} \frac{9}{11},\frac{6}{11},\frac{2}{11}\end{align}
Thus, the direction cosines are
\begin{align} \frac{9}{11},\frac{6}{11} and \frac{2}{11}\end{align}
Q4 Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear. Ans: The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7).
It is known that the direction ratios of line joining the points, (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}), are given by, x_{2} − x_{1}, y_{2} − y_{1}, and z_{2} − z_{1}.
The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., −3, −5, and −3.
The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., 6, 10, and 6.
It can be seen that the direction ratios of BC are −2 times that of AB i.e., they are proportional.
Therefore, AB is parallel to BC. Since point B is common to both AB and BC, points A, B, and C are collinear.
Q5 The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2). Ans: The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).
The direction ratios of side AB are (−1 − 3), (1 − 5), and (2 − (−4)) i.e., −4, −4, and 6.
Therefore, the direction cosines of AB are
The direction ratios of BC are (−5 − (−1)), (−5 − 1), and (−2 − 2) i.e., −4, −6, and −4.
Therefore, the direction cosines of BC are
The direction ratios of CA are (−5 − 3), (−5 − 5), and (−2 − (−4)) i.e., −8, −10, and 2.
Therefore, the direction cosines of AC are