FIND UNKNOWN IF THE GIVEN VECTORS ARE COPLANAR

Problem 1 :

If 

are coplanar. Find the value of m.

Solution :

Problem 2 :

The vectors

are coplanar if

a) 𝜆 = -2     b)  𝜆 = 0     c)  𝜆 = 2     d)  𝜆 = -1

Solution :

Since the given vectors are coplanar, while doing box product we will get 0.

Option a :

Applying 𝜆 = -2, we get

= (-2)³ - 6(-2) - 4

= -8 + 12 - 4

= -12 + 12

= 0

So, option a is correct.

Problem 3 :

The vectors

are coplanar, if m is equal to 

a)  1   b)  4   c)  3

d) no value of m for which it is coplanar.

Solution :

There is no variable m. So, option d is correct.

Problem 4 :

The number of distinct real values of 𝜆 for which the vectors

coplanar is

a) zero    b)  one     c) two    d) three

Solution :

-𝜆² [𝜆⁴ - 1] - 1[-𝜆² - 1] + 1[1 + 𝜆²] = 0

-𝜆⁶ + 𝜆² + 𝜆² + 1 + 1 + 𝜆² = 0

-𝜆⁶ + 3𝜆² + 2 = 0

𝜆⁶ - 3𝜆² - 2 = 0

(1+𝜆²)² (𝜆² - 2) = 0

𝜆² - 2 = 0

𝜆² = 2

𝜆 = 土√2

So, 𝜆 will have two distinct values.

Problem 5 :

If the vectors 

are coplanar, then the value of m is 

a)  5/8    b)  8/5     c)  -7/4      d)  2/3

Solution :

Since the given vectors are coplanar, its scalar triple product will be 0.

2[4 - 1] + 3[2 + m] + 4[-1 - 2m] = 0

2(3) + 3(2 + m) + 4(-1 - 2m) = 0

6 + 6 + 3m - 4 - 8m = 0

8 - 5m = 0

5m = 8

m = 8/5

Problem 6 :

If the vectors 

are coplanar, find the value of 𝜆.

Solution :

2[-𝜆 + 3] - 0[-𝜆 - 2] + 1[-3 - 2] = 0

-2𝜆 + 6 - 5 = 0

-2𝜆 + 1 = 0

-2𝜆 = -1

𝜆 = 1/2

So, the value of 𝜆 is 1/2.

Problem 1 :

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5,10)  lie on a same plane.

Solution :

Let the points  A = (6, -7, 0), B = (16, -19, -4), C = (0, 3, -6), D = (2, -5,10).

To show that the four points A, B, C and D lie on the plane, we have to prove that the three vectors 

Finding AB vector :

Finding AC vector :

Finding AD vector :

Proving the given four vectors are coplanar :

So, the given four vectors are coplanar.

Problem 2 :

Show that four points with position vectors

are coplanar.

Solution :

Since the given vectors are position vectors,

Finding AB vector :

Finding AC vector :

Finding AD vector :

Checking if the given position vectors are coplanar :

= -4[12+3] + 6[-3+24] - 2[1+32]

= -4[15] + 6[21] - 2[33]

= -60 + 126 - 66

= -126 + 126

= 0

So, the given points are coplanar.