Introduction To Three Dimensional Geometry

Let -plane = 0 divide join of and (1, 2, - 2) in the ratio K : 1 at the point P

Since this P lies on = 0 = = 0
K =
P =

Concept: 3D Geometry - Direction Ratios and Dot Product.

Step 1: Identify the geometric constraint. Since is right-angled at vertex A, the side vectors originating from A must be strictly perpendicular. Mathematically, this means the dot product of vector and vector must be exactly equal to zero ( ).

Step 2: Calculate vector . The vector from point to point is found by subtracting the coordinates of A from B: .

Step 3: Calculate vector . Similarly, the vector from point to point is: .

Step 4: Set up the dot product equation. Apply the condition . Multiply the corresponding , , and components and sum them: . This simplifies to .

Step 5: Solve the resulting quadratic equation. Combine the constant terms: . Factor the quadratic equation: . This gives two possible values for : or . Looking at the given options, only is available. $ $

Answer (d) 10

Answer (d) 7,10

To find the coordinates of the point where the line through points A(9, 4, 1) and B(5, 1, 6) crosses the X-axis, we need to determine the value of the X-coordinate when the Y and Z coordinates are both zero.

Step 1: Find the direction vector of the line:
The direction vector of the line can be found by subtracting the coordinates of B from A:

Step 2: Write the parametric equations for the line:
Using the direction vector, we can write the parametric equations for the line as: where is a parameter representing the position along the line.

Step 3: Set and to find the intersection with the X-axis:
The line crosses the X-axis when both and are zero. So, we solve the following system of equations:

Step 4: Solve the system of equations:
From the first equation, solve for : From the second equation, solve for :

Step 5: Determine the correct value of :
We have two different values of , and . However, since we are looking for the point where the line intersects the X-axis, we only need the value of that gives us the X-coordinate. Substituting the correct value of into the parametric equation for , we get:

Step 6: Final Answer:
Therefore, the coordinates of the point where the line crosses the X-axis are approximately: which simplifies to: