Questions: Three Dimensional Geometry

Q1 Define direction cosines and direction ratios for a line in space.

Answer (English): Take a directed line and the three coordinate axes. The cosines of the three angles between the line and the axes are its direction cosines, written l, m and n. They are the components of the unit vector along the line, so the sum of their squares is one. Direction ratios are any three numbers proportional to the direction cosines; the components of a vector parallel to the line are the simplest choice. From direction ratios a, b and c, the cosines are each ratio divided by the square root of a squared plus b squared plus c squared, with one common sign. Reversing the direction reverses all three cosines.

Hinglish: Direction cosines line ke unit vector ke components hote hain, isliye l²+m²+n² = 1. Direction ratios unke proportional numbers hote hain, aur vector ke components sabse aasan choice hain. Ratios se cosines nikalne ke liye √(a²+b²+c²) se divide karo aur sign common rakho. Direction ulti karne par teeno cosines ke signs badal jaate hain.

Q2 How is the equation of a line written in vector form and in cartesian form?

Answer (English): A line is fixed once one of its points and one direction along it are known. Take the point with position vector a and a parallel vector b; then every point of the line has position vector a plus lambda b for some real lambda, and that is the vector equation. In cartesian form the same line is (x - x1)/a1 = (y - y1)/a1 ... using the coordinates of the point and the direction ratios of b. The point locates the line and the direction orients it. If two points are given instead, the direction is the difference of their position vectors, and the cartesian form uses the differences of coordinates as denominators.

Hinglish: Vector equation r = a + λb hoti hai, jahan a point aur b direction hai. Cartesian form mein numerators se point aur denominators se direction ratios milte hain. Do points diye hon to direction b − a hoti hai.

Q3 How do you read a point and a direction from a cartesian equation?

Answer (English): In a cartesian equation of a line, the numbers subtracted in the numerators give the point, with their signs changed to read the coordinates. The denominators are the direction ratios. If a denominator is zero, the corresponding numerator must also be zero for points on the line, so the line lies in a plane of constant coordinate; this is a condition, not a division. The direction ratios may be multiplied by any non-zero scalar without changing the line.

Hinglish: Cartesian form mein numerators se point milta hai (sign ulta karke) aur denominators se direction ratios. Zero denominator ka matlab us coordinate ka numerator zero hoga, division nahi. Direction ratios ko kisi bhi non-zero scalar se multiply karne se line nahi badalti.

Q4 What is the formula for the angle between two lines?

Answer (English): The angle between two lines is the angle between their direction vectors. If the directions are b1 and b2, then the cosine of the acute angle is the modulus of b1 dot b2 divided by the product of their lengths. If the directions are given as direction ratios, the same formula is used with the ratios in place of components. Two lines are perpendicular when the dot product is zero, and parallel when the direction ratios are proportional, which is the same as a zero cross product.

Hinglish: Do lines ke beech ka angle unke direction vectors ka angle hota hai: cosθ = |b1·b2|/(|b1||b2|). Perpendicular par dot zero aur parallel par direction ratios proportional. Ratios diye hon to wahi formula components ki jagah ratios se.

Q5 How do you show that two lines are perpendicular or parallel?

Answer (English): For perpendicular lines, compute the dot product of the direction vectors and show that it is zero. For parallel lines, show that the three direction ratios are proportional, or that the cross product of the directions is the zero vector. A useful extra check is that the lines are not coincident: proportionality of directions proves parallelism regardless of position, but coincidence would need the point of one line to satisfy the equation of the other.

Hinglish: Perpendicular ke liye dot product zero dikhao; parallel ke liye direction ratios proportional ya cross product zero. Parallel prove karna position ke baare mein kuch nahi kehta, isliye coincident lines ke liye alag se point check karna padta hai.

Q6 How do you find the shortest distance between two skew lines?

Answer (English): Write the lines as r = a1 + lambda b1 and r = a2 + mu b2. The common perpendicular is along b1 cross b2, and the joining vector is a2 - a1. The shortest distance is the modulus of the triple product of the joining vector with the two directions, divided by the magnitude of the cross product. If the triple product is zero the lines are coplanar, so they either intersect or are parallel; in that case the distance is zero for intersecting lines and the parallel formula applies if the directions are parallel.

Hinglish: Skew lines ke liye shortest distance = |(a2−a1)·(b1×b2)|/|b1×b2|. Triple product zero matlab lines coplanar hain, isliye distance zero (intersecting) ya parallel formula use karo. Base points ka difference aur cross product dhyan se likho.

Q7 What is the distance between two parallel lines?

Answer (English): Parallel lines share one direction vector b, so write them as r = a1 + lambda b and r = a2 + mu b. The distance is the component of a2 - a1 perpendicular to b, which is the magnitude of (a2 - a1) cross b divided by the magnitude of b. The cross product gives the area of a parallelogram and dividing by the length turns it into a height. The same formula covers the case where one of the lines passes through the origin.

Hinglish: Parallel lines ke liye distance = |(a2−a1)×b|/|b|. Cross product area deta hai aur length se divide karne par height milti hai, jo shortest distance hai. Agar ek line origin se guzarti ho to bhi yahi formula lagta hai.

Q8 How do you find the foot of the perpendicular from a point to a line?

Answer (English): Let the line be r = a + lambda b and the point be P with position vector p. The foot F is the point on the line closest to P, so (p - a - lambda b) is perpendicular to b. Taking the dot product with b gives lambda equal to (p - a) dot b divided by the square of the length of b. Substituting this lambda gives the foot, and the distance from P to the line is the length of p - F, which also equals the magnitude of (p - a) cross b divided by the length of b.

Hinglish: Foot nikalne ke liye line r = a + λb lo aur point P ka vector p lo. Condition (p − a − λb)·b = 0 se λ = (p−a)·b/|b|² milta hai; λ daalo to foot mil jaata hai. Distance = |p − F| = |(p−a)×b|/|b|.

Q9 How do you check whether two lines intersect or are skew?

Answer (English): Write both lines with parameters and equate the coordinates. This gives three linear equations in two parameters. If a pair of parameters satisfies all three, the lines intersect at the corresponding point and the shortest distance is zero. If the equations are inconsistent, the lines do not meet. When the directions are not parallel and the lines do not meet, they are skew. The triple product test gives the same answer faster: zero means coplanar, so the lines meet or are parallel; non-zero means skew.

Hinglish: Dono lines ko parameter form mein likhkar coordinates barabar karo; teen equations aur do parameters milte hain. Consistent solution matlab lines intersect karti hain, aur inconsistent matlab nahi milti. Directions parallel nahi aur lines milti nahi to skew hain. Triple product zero = coplanar, non-zero = skew; yeh test tez hai.

Q10 How is the vector equation useful when a point and a perpendicular direction are given?

Answer (English): The cross product of two given directions gives a vector perpendicular to both, which is exactly the direction needed. Using the cross product of b1 and b2 as the direction, write r = a + lambda(b1 cross b2) for the line through the point a. Then convert to cartesian form using the components as direction ratios. If the question also asks for the distance from the origin, use the cross-product distance formula with a as the base point and the cross product as the direction.

Hinglish: Do directions ke cross product se perpendicular direction milti hai; line r = a + λ(b1×b2) ban jaati hai. Cartesian form ke liye cross product ke components denominators banti hain. Origin se distance chahiye to |a×d|/|d| formula use karo, jahan d = b1×b2 hai.

That completes the ten questions. The five MCQ quiz below checks the facts board papers test most often: the identity for direction cosines, the parallel test, the acute-angle formula, the skew-distance formula and the condition for zero distance.