The three vectors A = 3 i ^ − 2 j ^ + k ^ , B = i ^ − 3 j ^ + 5 k ^ and C = 2 i ^ − j ^ + 4 k ^ will form

Question

The three vectors A = 3 i ^ − 2 j ^ ​ + k ^ , B = i ^ − 3 j ^ ​ + 5 k ^ and C = 2 i ^ − j ^ ​ + 4 k ^ will form

TestHub · Accepted Answer

Three vectors form a triangle only if one of them is the sum or difference of the other two, so that the three vectors can be placed along the sides of the triangle.

$A + C = 5 \hat{i} - 3 \hat{j} + 5 \hat{k} \neq = B$

$B + C = 3 \hat{i} - 4 \hat{j} + 9 \hat{k} \neq = A$

$A - C = \hat{i} - \hat{j} - 3 \hat{k} \neq = \pm B$

In fact, the three vectors are not even coplanar:

$[A B C] = 312 - 2 - 3 - 1 154 = 3 (- 12 + 5) + 2 (4 - 10) + 1 (- 1 + 6) = - 21 - 12 + 5 = - 28 \neq = 0$

Although $∣ A ∣^{2} + ∣ C ∣^{2} = 14 + 21 = 35 = ∣ B ∣^{2}$ , the vectors cannot be joined end to end to close a triangle.

Therefore, the three vectors form no triangle.

Physics - Vectors Question with Solution | TestHub

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