IfA=2i-j-k,role="math" localid="1659148191947" B=2i-3j+k,find(A×B)C,,A(B×C),(A×B)×C,A×(B×C),(A×B)×C,A×(B×C).

For Problems 2 to 6, givenA=i+j-2k,B=2i-j+3k,C=j-5k

Short Answer

Expert verified

Simplifying, A=2i-j-k, B=2i-3j+k,C=j+kwe get

A×BC=6j+6kAB×C=-4i+2j+2k,A×B×C=A×B×C=-8A×B×C=4j-4kA×B×C=-4i-8k

Step by step solution

01

Definition of work done

The work is done by using cyclic permutations and using the identity of given parameters.

02

Given parameters

Now find the value of A×BC

A×BC=2i-j-k×2i-3j+kj+kA×BC=4+3-1j+kA×BC=6j+6k,1AB×C=2i-j-k2i-3j+k×j+kaAB×C=2i-j-k-3+1AB×C=-4i+2j+2k,2

03

Using cyclic permutations

Solving, A×B×C=A×B×Cusing the cyclic permutations method

A×B×C=2-1-1231011=2-3-1--12-0-12-0=-8

04

Using the identity:

Solving A×B×Cby identity method

A×B×C=A×CB-A×BCA×B×C=-C×A×B=-C×BA+C×AB

Substituting the values in the equation

C×AB=j+k×2i-j-k2i-3j+k)C×AB=-22i-3j+kC×AB=-4i+6j-2k),3

Solving localid="1659151677617" A×B×Cusing the identity method

A×B×C=--4i+2j+2k+-4i+6j-2k)A×B×C=4j-4k

Simplifying A×B×Cby identity method

A×B×C=-4i+6j-2k-6j+6kA×B×C=-4i+8k

Therefore, the work done as:

localid="1659152677024" A×BC=6j+6k,AB×C=-4i+2j+2k,A×B×C=A×B×C=-8A×B×C=4j-4kA×B×C=-4j-8k

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