The angle Q between the lines with direction cosines, a,b,c and b-c, c-a, a-b, is given by,
cosQ=|\(\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^2+b^2+c^2}+\sqrt{(b-c)^2+(c-a)^2+(a-b)^2}}\)|
⇒cosQ=0
⇒Q=cos-1=0
⇒Q=90°
Thus, the angle between the lines is 90°.
Show that the following lines intersect. Also, find their point of intersection:
Line 1: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \]
Line 2: \[ \frac{x - 4}{5} = \frac{y - 1}{2} = z \]
The vector equations of two lines are given as:
Line 1: \[ \vec{r}_1 = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(4\hat{i} + 6\hat{j} + 12\hat{k}) \]
Line 2: \[ \vec{r}_2 = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(6\hat{i} + 9\hat{j} + 18\hat{k}) \]
Determine whether the lines are parallel, intersecting, skew, or coincident. If they are not coincident, find the shortest distance between them.
Determine the vector equation of the line that passes through the point \( (1, 2, -3) \) and is perpendicular to both of the following lines:
\[ \frac{x - 8}{3} = \frac{y + 16}{7} = \frac{z - 10}{-16} \quad \text{and} \quad \frac{x - 15}{3} = \frac{y - 29}{-8} = \frac{z - 5}{-5} \]
Balance Sheet of Atharv and Anmol as at 31st March, 2024
Liabilities | Amount (₹) | Assets | Amount (₹) |
---|---|---|---|
Capitals: | Fixed Assets | 14,00,000 | |
Atharv | 8,00,000 | Stock | 4,90,000 |
Anmol | 4,00,000 | Debtors | 5,60,000 |
General Reserve | 3,50,000 | Cash | 10,000 |
Creditors | 9,10,000 | ||
Total | 24,60,000 | Total | 24,60,000 |
Column-I | Column-II |
---|---|
(a) Non-tax Revenue | (ii) Free-rider |
(b) Indirect Tax | (i) Goods and Services Tax |
(c) Capital expenditure | (iii) Borrowings |
(d) Private goods | (iv) Rivalrous in nature |