Geometrical applications MCQ Quiz in தமிழ் - Objective Question with Answer for Geometrical applications - இலவச PDF ஐப் பதிவிறக்கவும்

Calculation:

We have A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2)

∴ 

∴ Area of quadrilateral ABCD:

= 

= 

= 

= 

= 

= 9 sq. units

∴ The area of the quadrilateral ABCD is equal to 9 sq. units.

The correct answer is option 1

Given:

Three out of the four vertex of the parallelogram -

A = (1, 2, 3)

B = (-1, -2, -1)

C = (2, 3, 2)

Concept:

The diagonals of a paralleogram bisect each other.

Formula:

The mid-point of the line segment joining A (x1, y1, z1) and B (x2, y2, z2) is given by -

M (x, y, z) where ,  and 

Solution:

Let the midpoint of AC be M, then it is also the midpoint of BD

Now, M = (, , )

= (3/2, 5/2, 5/2)

Let D = (a, b, c)

Then, the midpoint of BD is given by (, , )

Equating it to M, we get a = 4, b = 7, c = 6

Hence D = (4, 7, 6)

Concept:

Triangle Law of Vector Addition: Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

Calculation:

In triangle ABC,

               …. (1)

In triangle BCD,

            …. (2)

Put the value of BC vector in equation 2^nd,

Concept:

Let A, B , and C be the vertices of the  , then the area of that triangle = 

Calculations:

Given, triangle whose vertices are at A = (3, -1, 2) = 

B = (1, -1, -3) =  

and C = (4, -3, 1) =  

Let A, B , and C be the vertices of the  , then the area of that triangle = 

⇒

⇒ 

Now, 

⇒

⇒

⇒ 

⇒

The area of that triangle = 

⇒ The area of that triangle = 
Hence, the area of a triangle whose vertices are at (3, -1, 2), (1, -1, -3) and (4, -3, 1) is 

Given

x = t² + 2, y = 4t - 5, z = 2t² - 6t

unit tangent vector is given as,

  = 4t - 6 = 4 × 2 - 6 = 2

Concept:

Calculation:

Let,  and 

And,

Hence, option (1) is correct.

Concept:

If the position vectors of points A and B are  and  respectively, then .

Area of a Triangle with vectors  and  as its sides is given by: .

Cross Product: For two vectors  and , their cross product is given by:

• .
• .
• .

The magnitude  of a vector  is given by: .

Calculation:

We have  and .

Using the formula for the area of a triangle:

= 

= 

= .

Additional Information

Area of a parallelogram with vectors  and  as its sides is given by: .

For two vectors  and  at an angle θ to each other:

• Dot Product is defined as: .
• Cross Product is defined as: , where  is the unit vector perpendicular to the plane containing  and .

Concept:

Calculation:

We have,    and 

Now, 

Area of triangle =  

= 1/2(5√6)

= ​​square unit

Hence, option (2) is correct.

Calculation:

By vector law,              

Similarly,    

Now, adding the above equations, we get 

From fig, AP = - BP and CP = -DP

∴  

= 4 OP 

 Hence, option (1) is correct.

Concept:

Area of rectangle ABCD = |AB| × |BC|

{Product of adjacent sides}

Calculation:

A =  B = 

⇒ AB = 

⇒ AB = 

⇒ |AB| = 2

Similarly,

B = , C = 

⇒ BC = 

⇒ BC = 

⇒ |BC| = 1

⇒ Area of rectangle ABCD = |AB| × |BC|

⇒ Area of rectangle ABCD = 2 × 1 = 2 sq unit

∴ The correct answer is option (3).