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Maths 3 Marks

Direction Cosines and Direction Ratios · Gujarat Board (GSEB) STD 12 Science (GSEB - English Medium). 7 questions.

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Question 13 Marks
Show that the line through points (1, -1, 2) and (3, 4, -2) is perpendicular to the line throught the points (0, 3, 2) and (3, 5, 6).
Answer
We know that two lines with direction ratios a1,b1, c2 and a2, b2, c2 are pependicular if a1a2 + b1b2 + c1c2 = 0.

The direction ratios of the line passing through the points (1, -1, 2) and (3, 4, -2) are (3 - 1), [4 - (-1)], (-2 - 2),

i.e. ⇒ a1 = 2, b1 = 2, c1 = -4

Similarly, the direction ratios of the line passing through the points (0, 3, 2) and (3, 5, 6) and (3 - 0), (5 - 3), (6 - 2),

i.e. ⇒ a2 = 3, b2 = 2, c2 = 4

$\therefore$ a1a2 + b1b2 + c1c2 = 2 × 3 + 5 × 2 (-4) × 4 = 6 + 10 - 16 = 0  

Thus the line through the points (1, -1, 2) and (3, 4, -2) is perpendicular to the line throught the points (0, 3, 2) and (3, 5, 6).

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Question 23 Marks
Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).
Answer
The direction consines of the line passing through two points P x1, y1, z1, and Q (x2, y2, z2) are $\frac{\text{x}_2-\text{x}_1}{\text{PQ}},\frac{\text{y}_2-\text{y}_1}{\text{PQ}},\frac{\text{z}_2-\text{z}_1}{\text{PQ}}.$
Here,
$\text{PQ}=\sqrt{(\text{x}_2-\text{x}_1)^2+(\text{y}_2-\text{y}_1)^2+(\text{z}_2-\text{z}_1)^2}$
$\text{P}=2,4,-5$
$\text{Q}=1,2,3$
$\therefore\text{PQ}=1-(-2)^2+(2-4)^2+[3-(-5)]^2=\sqrt{77}$
Thus, the direction cosines of the line joining two points are
$\frac{1-(-2)}{\sqrt{77}},\frac{2-4}{\sqrt{77}},\frac{3-(-5)}{\sqrt{77}},\text{i.e.}\frac{3}{\sqrt{77}}77,\frac{-2}{\sqrt{77}}77,\frac{8}{\sqrt{77}}.$
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Question 33 Marks
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1) and (4, 3, -1).
Answer
Here,
A(0, 0, 0) and B(2, 1, 1)
C(3, 5, -1) and D(4, 3, -1)
Direction ratios of line AB
a1 = 2, b1 = 1, c1 = 1
Direction ratios of line CD
a2 = 2, b2 = -2, c2 = 0
Now,
a1a2 + b1b2 + c1c2
= (2)(1) + (1)(-2) + (1)(0)
= 2 - 2 + 0
= 0
Since, a1a2 + b1b2 + c1c2 = 0, lines are perpendicular.
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Question 43 Marks
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line throught the points (-1, -2, 1) and (1, 2, 5).
Answer
Suppose the points are A(2, 3, 4), B(-1, -2, 1) and C(5, 8, 7).
We know that the direction ratios of the line passing through the points (x1, y1, z1) and (x2, y2, z2) are x2 - x1, y2 - y1, z2 - z1.
Let the first two points be A(4, 7, 8) and B(2, 3, 4).
Thus, the direction ratios of AB are (2 - 4), (3 - 7), (4 - 8), i.e. -2, -4, -4.
Similarly, Let the other two points be C (-1, -2, 1) and D (1, 2, 5).
Thus, the direction ratios of CD are [1 - (-1)], [2 - (-2)], (5 - 1), i.e. 2, 4, 4.
It can be seen that the direction ratios of CD are -1 times that of AB, i.e. they are proportional. Therefore, AB and CD are parallel lines.
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Question 53 Marks
Write the angle between the lines whose direction ratios are perportional to 1, -2, 1 and 4, 3, 2.
Answer
The direction ratios of the first line are 1, -2, 1 and the direction ratios of the second line are 4, 3, 2.
Let $\theta$ be the angle between these two lines.
Now,
$\cos\theta =\Bigg|\frac{1(4)+(-2)(3)+1(2)}{\sqrt{(1)^2+(-2)^2+(1)^2}\sqrt{(4)^2+(3)^2+(2)^2}}\Bigg|$
$=\Bigg|\frac{4+6+2}{\sqrt{1+4+1}\sqrt{16+9+4}}\Bigg|$
$=\frac{0}{\sqrt{6}\sqrt{29}}$
$=0$
$\Rightarrow\theta=\frac{\pi}{2}$
Hence, the required angle is$​​​​\frac{\pi}{2}$.
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Question 63 Marks
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
Answer
It is given that a line makes an angle of 60° with both x-axis and y-axis.
Suppose the line makes an angle of $\alpha$ with the z-axis.
$\Rightarrow\text{l}=\cos60^\circ=\frac{1}{2}\text{m}$
$=\cos60^\circ=\frac{1}{2}\text{n}=\cos\alpha$
We know $\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\Big(\frac{1}{2}\Big)^2+\Big(\frac{1}{2}\Big)^2+(\cos\alpha)^2=1$
$\Rightarrow\frac{1}{4}+\frac{1}{4}+\cos ^2\alpha=1$
$\Rightarrow\cos\alpha=\frac{1}{\sqrt{2}}$
$\Rightarrow\alpha=45^\circ$
Thus, the line makes an angle of 45° with the z-axis.
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Question 73 Marks
If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}{3}$ with $\hat{\text{i}},\frac{\pi}{4}$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, and ,then find the value of $\theta$.
Answer
Scince a unit vector makes an angle of $\frac{\pi}{3}$ with$\hat{\text{i}}$, $\frac{\pi}{4}$ with $\hat{\text{j}}$ andan acute angle $\theta$ with $\hat{\text{k}},\text{l}=\cos\frac{\pi}{3}$ or $\frac{\pi}{4}$ or $\frac{1}{\sqrt{2}}$and $\text{n}=\cos\theta$.
We know
$\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\Big(\frac{1}{2}\Big)^2+\Big(\frac{1}{\sqrt{2}}\Big)^2+\cos^2\theta=1$
$\Rightarrow\frac{1}{4}+\frac{1}{2}+\cos^2\theta$
$\Rightarrow\cos^2\theta=\frac{1}{4}$
$\Rightarrow\cos^2\theta=\frac{1}{2}$
$\Rightarrow\frac{\pi}{3}$
Thus, the vector $\vec{\text{a}}$ makes an angle of $\frac{\pi}{3}$ with $\hat{\text{k}}$.
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