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Vector Equations and Line Formulas Quiz

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Questions and Answers

What is the vector equation of a straight line passing through two points with position vectors a and b?

r = a + λ(b - a) (correct)
r = a - λ(b - a)
r = a - λ(b + a)
r = a + λ(b + a)

Given the equation of a line in Cartesian form as z = c, what is its corresponding vector form?

r = c(k - i)
r = c(k + i)
r = c(k - k1) (correct)
r = c(k + k1)

If two lines are coplanar, what is the condition expressed by the cross product of their direction vectors?

The cross product is normal to the lines.
The cross product is parallel to the plane containing the lines.
The cross product is perpendicular to the plane containing the lines. (correct)
The cross product is tangent to the lines.

What is the vector equation of a line through the origin with direction cosines (l, m, n)?

r = λ(l i + m j + n k) (C) Signup and view all the answers

In the context of intersecting lines, what does the variable T represent?

Angle between the lines (C) Signup and view all the answers

For lines given by ax + by + cz + d = 0 and a'x + b'y + c'z + d' = 0 to be coplanar, what condition must be satisfied?

$|a b c| = |a' b' c'|$ (C) Signup and view all the answers

What is the vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b?

r = a + λb (C) Signup and view all the answers

What is the equation of a line passing through the points (x1, y1, z1) and (x2, y2, z2) in vector form?

(x - x2) / (x1 - x2) = (y - y1) / (y2 - y1) = (z - z2) / (z1 - z2) (C) Signup and view all the answers

Which type of form characterizes a straight line in space by the intersection of two non-parallel planes?

Non-symmetric form (B) Signup and view all the answers

How is a general point on a line defined with direction ratios a, b, c and passing through the point (x1, y1, z1)?

(x1 + ar, y1 + br, z1 + cr) (B) Signup and view all the answers

What is the equation of any plane passing through a given line in symmetrical form?

(r ⋅ n₂) = d₁ (C) Signup and view all the answers

For the equation of a line in space defined by two intersecting planes, which condition must the coefficients satisfy?

|a₁b₂ − a₂b₁| ≠ 0 (D) Signup and view all the answers

If a plane is parallel to the x-axis, what is the equation of the plane?

by + cz + d = 0 (D) Signup and view all the answers

What is the equation of a plane cutting intercepts a, b, c on the axes?

a/x + b/y + c/z = 1 (B) Signup and view all the answers

What is the vector form of the projection of a vector a on another vector b?

|a| (l î + m ĵ + n k̂) (C) Signup and view all the answers

Which equation represents the yz-plane?

x = 0 (D) Signup and view all the answers

What does the equation r n ˆ d represent in the context of a plane?

Vector equation of a plane normal to unit vector n at distance d from the origin (B) Signup and view all the answers

Which form of the projection formula in vectors involves l, m, and n?

| l (x2 – x1) + m(y2 – y1) + n(z2 – z1) | (D) Signup and view all the answers

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Study Notes

Vector and 3D Geometry

Equation of a line in Cartesian form: a1x + b1y + c1z + d1 + O(a2x + b2y + c2z + d2) = 0
Vector equation of a straight line passing through two points with position vectors a and b: r = a + O(b - a)
Reduction of Cartesian form of equation of a line to vector form: x - x1 = a, y - y1 = b, z - z1 = c and r = (x1, y1, z1) + O(ai + bj + ck)

Coplanar Lines

Condition for intersection of two lines: A / m' = B / n' = C / n
Condition of coplanarity of two lines: aA + bm + cn sin T = 0
Coplanarity of two lines in general asymmetric form: ax + by + cz + d = 0 and a'x + b'y + c'z + d' = 0

Projection of a Line Segment on a Line

Projection of a line segment PQ on a line with direction cosines l, m, n: |l(x2 - x1) + m(y2 - y1) + n(z2 - z1)|
Vector form: a ∘ b is the projection of vector a on another vector b

Plane

Intercept form: x/a + y/b + z/c = 1
Vector form: r ∘ n = 0 or r ∘ n = a ∘ n
Equation of a plane normal to unit vector n and at a distance d from the origin: r ∘ n = d
Planes parallel to the coordinate planes: x = 0, y = 0, z = 0
Planes parallel to the axes: by + cz + d = 0, bx + cz + d = 0, bx + cy + d = 0

Area of a Triangle

Area of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3): |(x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1)|

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