Question:

Find the distance between the following pairs of points: (i) (2,3,5) and (4,3,1) (ii)(−3,7,2)and ((2,4,−1) (iii) (−1,3,−4) and (1,−3,4) (iv) (2,−1,3) and (−2,1,3)

Answer:

(i) Distance = √((4-2)² + (3-3)² + (1-5)²) = √(4+0+16) = √20 = 4.4721

(ii) Distance = √((2-(-3))² + (4-7)² + ((-1)-2)²) = √(5+9+9) = √23 = 4.7958

(iii) Distance = √((1-(-1))² + ((-3)-3)² + (4-(-4))²) = √(2+36+16) = √54 = 7.3484

(iv) Distance = √((-2-2)² + (1-(-1))² + (3-3)²) = √(4+2+0) = √6 = 2.4495

Question:

Show that the points (−2,3,5),(1,2,3) and (7,0,−1) are collinear.

Answer:

Step 1: Calculate the equation of the line passing through the points (−2,3,5) and (1,2,3).

Let the equation of the line passing through the points (−2,3,5) and (1,2,3) be: ax + by + cz = d

Substituting the coordinates of the points (−2,3,5) and (1,2,3) in the above equation, we get: -2a + 3b + 5c = d a + 2b + 3c = d

Solving the above equations, we get: a = -3, b = 5, c = -2, d = 13

Step 2: Substitute the coordinates of the point (7,0,-1) in the equation of the line obtained in Step 1.

Substituting the coordinates of the point (7,0,-1) in the equation of the line obtained in Step 1, we get: -3(7) + 5(0) + (-2)(-1) = 13 -21 - 2 = 13

Step 3: Check if the equation obtained in Step 2 is true.

The equation obtained in Step 2 is -23 = 13 which is false.

Therefore, the points (−2,3,5),(1,2,3) and (7,0,−1) are not collinear.

Question:

Verify the following (i) (0,7,−10),(1,6,−6) and (4,9,−6) are the vertices of an isosceles triangle (ii) (0,7,10),(−1,6,6) and (−4,9,6) are the vertices of a right angled triangle (iii) (−1,2,1),(1,−2,5),(4,−7,8) and (2,−3,4) are the vertices of a parallelogram

Answer:

(i) To verify that (0,7,−10),(1,6,−6) and (4,9,−6) are the vertices of an isosceles triangle, we need to check if two of the sides are equal.

Let AB = (0,7,−10) and (1,6,−6) and AC = (0,7,−10) and (4,9,−6).

We calculate the length of AB and AC using the distance formula.

AB = √((1-0)²+(6-7)²+(−6-(−10))²) = √(1+1+16) = √18

AC = √((4-0)²+(9-7)²+(−6-(−10))²) = √(16+4+16) = √36

Since AB = AC, the triangle is isosceles.

(ii) To verify that (0,7,10),(−1,6,6) and (−4,9,6) are the vertices of a right angled triangle, we need to check if one of the angles is 90°.

We calculate the lengths of the sides using the distance formula.

AB = √((−1-0)²+(6-7)²+(6-10)²) = √(1+1+16) = √18

AC = √((−4-0)²+(9-7)²+(6-10)²) = √(16+4+16) = √36

BC = √((−4-(−1))²+(9-6)²+(6-6)²) = √(9+9+0) = √18

We then use the Pythagorean theorem to calculate the angle.

AB² + AC² = BC²

18² + 36² = 18²

324 + 1296 = 324

Therefore, the angle between AB and AC is 90°, so the triangle is right angled.

(iii) To verify that (−1,2,1),(1,−2,5),(4,−7,8) and (2,−3,4) are the vertices of a parallelogram, we need to check if opposite sides are equal.

Let AB = (−1,2,1) and (1,−2,5) and CD = (1,−2,5) and (4,−7,8).

We calculate the length of AB and CD using the distance formula.

AB = √((1-(−1))²+((−2)-2)²+((5-1))²) = √(4+16+16) = √36

CD = √((4-(1))²+((−7)-(−2))²+((8-5))²) = √(9+25+9) = √43

Since AB = CD, the quadrilateral is a parallelogram.

Question:

Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,−1)

Answer:

Answer: Step 1: Find the mid-point of the two given points. Mid-point = (1+3/2, 2+2/2, 3+(-1)/2) = (2,2,1)

Step 2: Find the distance between the two points. Distance = √((3-1)² + (2-2)² + (-1-3)²) = √12

Step 3: Write the equation of a sphere with radius = distance and center = mid-point. Equation of the sphere = (x-2)² + (y-2)² + (z-1)² = 12

Question:

Find the equation of the set of points P, the sum of whose distances from A(4,0,0) and B(−4,0,0) is equal to 10.

Answer:

1. Find the distances from A(4,0,0) and B(-4,0,0) to any point P(x,y,z).

Distance from A to P: dA = √((x-4)² + y² + z²) Distance from B to P: dB = √((x+4)² + y² + z²)

2. Set the sum of the distances equal to 10: dA + dB = 10

3. Simplify the equation: √((x-4)² + y² + z²) + √((x+4)² + y² + z²) = 10

4. Square both sides of the equation: (x-4)² + y² + z² + (x+4)² + y² + z² = 100

5. Simplify the equation: 2x² + 8 = 100

6. Solve for x²: x² = 46

7. Take the square root of both sides: x = ±√46

8. Therefore, the equation of the set of points P is x = ±√46, y, z.