Chapter: Analytical Geometry of Two & Three-Dimensions
Consider the following statements :
1. The direction ratios of y-axis can be < 0, 4, 0 >
2 . The direction ratios of a line perpendicular to z-axis can be < 5, 6, 0 >.
Which of the statements given above is /are correct ?
Solution:
Statement 1:
Direction cosines of y-axis:
⟨0, 1, 0⟩
Direction ratios can be proportional.
⟨0, 4, 0⟩ is valid.
Statement 1 is correct.
Statement 2:
Direction cosines of z-axis:
⟨0, 0, 1⟩
For perpendicular lines:
a₁a₂ + b₁b₂ + c₁c₂ = 0
Checking ⟨5, 6, 0⟩ with z-axis:
5(0) + 6(0) + 0(1)
= 0
Therefore line is perpendicular to z-axis.
Statement 2 is correct.
Hence option (c).
If l, m, n are the direction cosines of the line x − 1 = 2(y + 3) = 1 − z, then what is l⁴ + m⁴ + n⁴ equal to?
What is the angle between the two lines having direction ratios ⟨6, 3, 6⟩ and ⟨3, 3, 0⟩?
For the next two (02) items that follow:
A line L passing through the point (−1, 2, −3) is perpendicular to the plane P given by
2x + 3y + z + 5 = 0.
What are the direction ratios of a line M parallel to the plane P?
Option (d) is correct.
Explanation:
The direction ratios of the normal to the plane \( P: 2x + 3y + z + 5 = 0 \) are given by the coefficients of \( x, y, z \). \[ \text{Normal Direction Ratios: } \langle 2, 3, 1 \rangle \] A line \( M \) that is parallel to the plane \( P \) must have a direction vector that is orthogonal to the normal vector. If the direction ratios of the line \( M \) are \( \langle a, b, c \rangle \), then the dot product must equal zero: \[ 2a + 3b + c = 0 \] Testing the options: - For option A: \(\langle -3, 2, 1 \rangle\) \[ 2(-3) + 3(2) + 1 = -6 + 6 + 1 = 1 \quad (\text{not zero}) \] - For option B: \(\langle 3, 2, -6 \rangle\) \[ 2(3) + 3(2) - 6 = 6 + 6 - 6 = 6 \quad (\text{not zero}) \] - For option C: \(\langle 1, 3, 2 \rangle\) \[ 2(1) + 3(3) + 2 = 2 + 9 + 2 = 13 \quad (\text{not zero}) \] - For option D: \(\langle 2, 2, -10 \rangle\) \[ 2(2) + 3(2) - 10 = 4 + 6 - 10 = 0 \quad (\text{is zero}) \] Hence, option (d) is indeed the correct choice. The direction ratios \( \langle 2, 2, -10 \rangle \) align with the required orthogonality.A coordinate system where the plane \( P \) is represented as a surface, and the normal vector is depicted as an arrow perpendicular to this surface. The direction vectors \( \langle a, b, c \rangle \) should illustrate how they lie in the plane, confirming they are orthogonal to the normal.
Verified against standard texts on Analytical Geometry and Vector Algebra.
For the following two (02) items:
A plane P is parallel to the line having direction ra- tios ⟨1, 3, 2⟩ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0.
Which of the following are the direction ratios of the line of intersection of the given planes?
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