Question

If a = 7i - 2j + 3k, b = 2i +8k and c = i + j + k then compute a × b, a × c and a × (b + c). Verify whether the cross product is distributive over vector addition.

Answer 1

Answer:

Yes, The Cross Product is Distributive Over Vector Addition.

Step-by-step explanation:

The given vectors are

a = 7i - 2j + 3k

b = 2i +8k

c = i + j + k

First, we need to compute the vector product of the given vectors as asked in question.

Then, we need to show that Cross Product is Distributive over Vector Addition.

I have solved in detail.

Kindly see the attachment for the detailed answer.

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Answer 2

The cross product of a, b and c is distributive over vector addition.

Given:

a = 7i - 2j + 3k

b = 2i + 8k

c = i + j + k

To Find:

• a × b
• a × c
• a × (b + c)

If cross product is distributive over vector addition

Solution:

To verify the statement we must find the following :

1. a × b

a = 7i - 2j + 3k      b = 2i + 8k

• Here, we will be using the matrix method to find the cross product of the two vectors

                     

    $a * b = \ \ \left[\begin{array}{ccc}i&j&k\\7&-2&3\\2&0&8\end{array}\right]$

$a * b = i[(8*-2) - (0*3)] - j[(7*8)-(2*3)] +k[(7*0)-(2*-2)]\\\\a * b = i(-16-0) -j(56-6) +k(0-(-4))\\\\a * b = -16i -50j +4k$

2. a × c

• Matrix method to be used to solve the following cross product

      $a * c = \left[\begin{array}{ccc}i&j&k\\7&-2&3\\1&1&1\end{array}\right]$

$a * c = i[(-2*1)-(3*1)] -j[(7*1)-(1*3)] +k[(7*1)-(1*-2)]\\\\a*c = i(-2 - 3) -j(7-3) +k(7+2)\\\\a*c = -5i -4j +9k$

3. (a × b) + (a × c)

a × b = -16i -50j -4k

a × c = -5i -4j + 9k

$(a * b) + ( a *c) =\\\\i(-16 + -5) + j(-50 + -4) +k( 4 +9)\\\\(a*b) + (a*c) = -21i -54j + 13k$      - Eq A

4. a × (b + c)

• First, solve the b + c part of the vector problem:

$b + c = [2i + 0j + 8k] + [i + j + k]\\\\ = i(2 +1) + j(0 +1) + k(8 +1)\\ \\ = 3i + j + 9i$

• Now, find the cross product of a and (b + c)

$a * (b+c) = \left[\begin{array}{ccc}i&j&k\\7&-2&3\\3&1&9\end{array}\right]$

$a *(b+ c) = i[(-2*9)-(3*1)] -j[(7*9)-(3*3)] +k[(7*1)-(3*-2)]\\\\a*(b+c)= i(-18 - 3) -j(63-9) +k(7+6)\\\\a*(b+c) = -21i -54j +13k$

• We see that,

        (a × b) + (a × c) ≡ a × (b + c) ≡ -21i - 54j + 13k

        Hence, the cross product is distributive over vector addition.

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