PERPENDICULAR VECTORS WORKSHEET

1. Check whether the following vectors are perpendicular to each other. 

(4i + 2j - 8k) and (3i - 2j + k)

2. Check whether the following vectors are perpendicular to each other. 

(i - 5j + 7k) and (-2i + 3j +4k)

3. Show that the vectors a = 2i + 3j + 6k, b = 6i + 2j − 3k, and c = 3i − 6j + 2k are mutually orthogonal.

4. Let -i - 2j - 6k,  2i - j + k, and -i + 3j + 5k be the sides of a triangle. Show that the above vectors form a right triangle.

5. If the vectors (i + 2j - 5k) and (3i - 2j + ak) are perpendicular to each other, find the value of 'a'.

1. Answer :

Find the dot product of the given two vectors :

= (4i + 2j - 8k)  (3i - 2j + k)

= 4(3) + 2(-2) - 8(1)

= 12 - 4 - 8

= 0

Since the dot product is equal to zero, the given two vectors are perpendicular to each other. 

2. Answer :

Find the dot product of the given two vectors : 

= (i - 5j + 7k)  (-2i + 3j + 4k)

= 1(-2) - 5(3) + 7(4)

= -2 - 15 + 28

= 11 ≠ 0

Since the dot product is not equal to zero, the given two vectors are not perpendicular to each other. 

3. Answer :

Mutually orthogonal means, they are perpendicular to each other.

a vector  b vector = (2i + 3j + 6k)  (6i + 2j − 3k)

= 2(6) + 3(2) + 6(-3)

= 12 + 6 - 18

= 18 - 18

= 0

b vector  c vector = (6i + 2j − 3k)  (3i − 6j + 2k)

= 6(3) + 2(-6) + (-3)2

= 18 - 12 - 6

= 18 - 18

= 0

c vector  a vector = (3i − 6j + 2k (2i + 3j + 6k)

= 3(2) + (-6)(3) + 2(6)

= 6 - 18 + 12

= 18 - 18

= 0

Since, a ⋅ b = b ⋅ c = c ⋅ a = 0, the given vectors are mutually orthogonal.

4. Answer :

The given vectors are the sides of a right triangle. To show that the triangle is a right triangle, it is enough to prove that the two of its sides are perpendicular. 

Let

a = -i - 2j - 6k

b = 2i - j + k 

c = = -i + 3j + 5k

 b = (-i - 2j - 6k)  (2i - j + k)

= -1(2) + (-2)(-1) + (-6)(1)

= -2 + 2 - 6

= -6 ≠ 0

⋅ c = (2i - j + k (-i + 3j + 5k)

  = 2(-1) + (-1)(3) + 1(5)

= -2 - 3 + 5

= 0

Since b ⋅ c = 0, b vector and c vector are perpendicular.

So, the given vectors form a right triangle. 

5. Answer :

Since , the given two vectors are perpendicular, their dot product is equal to zero. 

(i + 2j - 5k)  (3i - 2j + ak) = 0

1(3) + 2(-2) - 5(a) = 0

3 - 4 - 5a = 0

-1 - 5a = 0

a = -1/5

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Trigonometry Word Problems Worksheet with Answers

    Jan 17, 22 10:45 AM

    Trigonometry Word Problems Worksheet with Answers

    Read More

  2. Trigonometry Word Problems with Solutions

    Jan 17, 22 10:41 AM

    Trigonometry Word Problems with Solutions

    Read More

  3. Writing Numbers in Words Worksheet

    Jan 16, 22 11:56 PM

    Writing Numbers in Words Worksheet

    Read More