Questions: Vector Algebra

Q1 Define a vector and explain the main types of vectors.

Answer (English): A vector is an object with both size and direction, drawn as a directed line segment. The zero vector has zero magnitude and no fixed direction. A unit vector has length one and is written with a hat. Coinitial vectors share an initial point. Collinear vectors are parallel to one line, regardless of length. Equal vectors have the same magnitude and direction, and the negative of a vector has the same magnitude with the opposite direction. A free vector may be shifted parallel to itself without change.

Hinglish: Vector ke paas magnitude aur direction dono hote hain aur use directed line segment se dikhate hain. Zero vector ki magnitude zero hoti hai, unit vector ki length ek, coinitial vectors ka initial point same hota hai, aur collinear vectors ek hi line ke parallel hote hain. Equal vectors ka magnitude aur direction same hote hain, aur negative vector ki direction ulti hoti hai. Free vector ko parallel shift karne se kuch nahi badalta.

Q2 State the triangle and parallelogram laws of addition and prove that addition is commutative.

Answer (English): The triangle law says that if two vectors are represented by two sides of a triangle taken in order, the third side taken from the start of the first to the end of the second represents the sum. Equivalently, the parallelogram law says that when the two vectors are adjacent sides of a parallelogram, the diagonal through the common point is the sum. For commutativity, draw a parallelogram ABCD with AB = a and AD = b. By the triangle law on triangle ABC, AC = a + b. By the triangle law on triangle ADC, AC = b + a. Both give the same diagonal, so a + b = b + a.

Hinglish: Triangle law mein do vectors ek triangle ki do sides banate hain aur teesri side sum hoti hai. Parallelogram law mein dono vectors common point se adjacent sides bante hain aur diagonal sum deta hai. Commutativity ke liye parallelogram ABCD lo: triangle ABC se AC = a + b, triangle ADC se AC = b + a, dono same diagonal hain, isliye a + b = b + a.

Q3 Write a vector in component form, find its magnitude and its unit vector.

Answer (English): With unit vectors i, j and k along the axes, a vector is written as x i + y j + z k. Its magnitude is the square root of x squared plus y squared plus z squared. The unit vector in its direction is the vector divided by that magnitude. For a = 2 i + 3 j + 6 k, the magnitude is the square root of 4 + 9 + 36, which is 7, so the unit vector is (2 i + 3 j + 6 k) divided by 7. The unit vector carries the direction and has length one.

Hinglish: Component form x i + y j + z k hota hai, magnitude √(x²+y²+z²) hoti hai, aur unit vector vector ko magnitude se divide karke milta hai. a = 2 i + 3 j + 6 k ki length 7 hai, isliye unit vector (2 i + 3 j + 6 k)/7 hai. Unit vector ki length hamesha ek hoti hai.

Q4 Explain direction cosines and direction ratios, and give their relation.

Answer (English): If a vector makes angles alpha, beta and gamma with the positive coordinate axes, then cos alpha, cos beta and cos gamma are its direction cosines l, m and n. They are the components of the unit vector in that direction, so l squared plus m squared plus n squared equals one. Direction ratios are any three numbers proportional to the direction cosines; the components of the vector are the simplest choice. For direction ratios a, b, c, the direction cosines are a, b and c each divided by the square root of a squared plus b squared plus c squared, with one common sign.

Hinglish: Direction cosines cos α, cos β, cos γ hote hain aur unit vector ke components hote hain, isliye l²+m²+n² = 1. Direction ratios unke proportional koi bhi numbers hote hain; vector ke components sabse aasan choice hain. Direction ratios a, b, c se cosines nikalne ke liye har ratio ko √(a²+b²+c²) se divide karo aur ek common sign rakho.

Q5 State and explain the section formula for internal and external division.

Answer (English): Let A and B have position vectors a and b. The point R that divides AB internally in the ratio m : n has position vector (m b + n a) divided by (m + n). The point that divides it externally in the same ratio has position vector (m b - n a) divided by (m - n). For internal division both segments are on the same side of the point, so the weighted vectors add; for external division the point lies beyond one end, so they subtract. Putting m equal to n in the internal formula gives the midpoint, (a + b) divided by 2.

Hinglish: Section formula mein internal division ke liye r = (m b + n a)/(m + n) aur external division ke liye r = (m b − n a)/(m − n) hota hai. Internal mein parts same side hote hain isliye add karte hain, external mein point bahar hota hai isliye subtract karte hain. m = n rakho to midpoint (a + b)/2 milta hai.

Q6 Define the dot product and use it to find the angle between two vectors.

Answer (English): The dot product of two nonzero vectors is the product of their magnitudes and the cosine of the angle between them, and it also equals the sum of the products of corresponding components. The result is a scalar. Rearranging gives cos theta equals a dot b divided by the product of the magnitudes. For a = i - j + k and b = i + j - k, the dot product is -1 and each magnitude is root 3, so the cosine is minus one third, which gives an obtuse angle. Two nonzero vectors are perpendicular exactly when their dot product is zero.

Hinglish: Dot product |a||b|cosθ ke barabar hai aur components se a1b1+a2b2+a3b3 bhi hota hai; result scalar hota hai. Angle ke liye cosθ = a·b/(|a||b|). Example mein dot product −1 aur dono lengths √3 hain, isliye cosθ = −1/3 aur angle obtuse hai. Perpendicular vectors ka dot product zero hota hai.

Q7 How do you find the projection of one vector on another?

Answer (English): The scalar projection of a on b is the dot product of a and b divided by the magnitude of b. It is the signed length of the part of a along the direction of b, so it is positive for an acute angle, zero when the vectors are perpendicular and negative for an obtuse angle. For a = 2 i + 3 j + 2 k and b = i + 2 j + k, the dot product is 10 and the magnitude of b is root 6, so the projection is 10 divided by root 6.

Hinglish: Projection scalar hoti hai aur a·b/|b| se milti hai; ye b ki direction mein a ka signed length hai. Acute angle par positive, perpendicular par zero aur obtuse par negative hoti hai. Example mein dot product 10 aur |b| = √6 hai, isliye projection 10/√6 hai. Vector projection ke liye is scalar ko b ke unit vector se multiply karte hain.

Q8 Define the cross product and give its determinant form.

Answer (English): The cross product of nonzero vectors a and b is a vector perpendicular to both, with magnitude equal to the product of their lengths and the sine of the angle between them. Its direction follows the right hand rule, which makes it non-commutative: b cross a equals minus a cross b. In components, write i, j, k in the first row, the components of a in the second and of b in the third, and expand the determinant, remembering the minus sign on the j term. The coordinate unit vectors give i cross j = k, j cross k = i and k cross i = j.

Hinglish: Cross product dono vectors ke perpendicular vector deta hai, magnitude |a||b|sinθ aur direction right hand rule se. Ye non-commutative hai: b×a = −a×b. Component form determinant se, aur j wale term ka minus sign yaad rakho. Parallel vectors ka cross product zero hota hai.

Q9 Give the formulas for the area of a triangle and a parallelogram using vectors.

Answer (English): If a and b are two sides drawn from a common vertex, the area of the parallelogram they span is the magnitude of a cross b, and the area of the triangle on the same sides is half of that magnitude. For A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1), the sides AB = j + 2 k and AC = i + 2 j give a cross product of -4 i + 2 j - k, whose magnitude is root 21. The area of triangle ABC is therefore half of root 21.

Hinglish: Agar do sides a aur b ek common vertex se nikalti hain, to parallelogram ka area |a×b| hota hai aur triangle ka area ½|a×b|. Example mein AB = j + 2k aur AC = i + 2j ka cross product −4i + 2j − k hai jiski magnitude √21 hai, isliye triangle ka area √21/2 hai. Area ka answer hamesha positive hota hai, isliye half karne se pehle modulus lo.

Q10 Give one example each of a collinearity proof and a perpendicular vector construction.

Answer (English): For collinearity, form the vectors from the first point to the second and from the second to the third. If one is a scalar multiple of the other, the three points lie on one line; for A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1), the two joining vectors are both i + 4 j - 4 k, so the points are collinear. For a perpendicular direction, take the cross product of the two given vectors and divide by its magnitude, which gives the unit vectors normal to their plane.

Hinglish: Collinearity ke liye do joining vectors nikalo; agar ek doosre ka scalar multiple ho to points ek line par hain. Perpendicular unit vector ke liye cross product ko uski magnitude se divide karo, phir ± sign ke saath likho.

That completes the ten questions. The five MCQ quiz below checks the unit vector formula, the perpendicularity test, the projection, the cross product magnitude and the triangle area.