Q17P Find three vectors (none of them... [FREE SOLUTION]

Chapter 3: Q17P (page 105)

Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Short Answer

Expert verified

The three vectors are $2 \hat{i} + \hat{j} - \hat{k}$ , $\hat{i} + 3 \hat{j} + 5 \hat{k}$ and $3 \hat{i} + 4 \hat{j} + 4 \hat{k}$ .

Step by step solution

Required

Three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Results used

Vector $\overset{鈫�}{A} 脳 \overset{鈫�}{B}$ is perpendicular to vector $\overset{鈫�}{A}$ as well as $\overset{鈫�}{B}$ .

Consider two vectors and find a perpendicular vector

Consider two vectors,

$\overset{鈫�}{A} = 2 \hat{i} + \hat{j} - \hat{k}$

$\overset{鈫�}{B} = \hat{i} + 3 \hat{j} - 2 \hat{k}$

$\overset{鈫�}{A} 脳 \overset{鈫�}{B} = |\begin{matrix} i & j & k \\ 2 & 1 & - 1 \\ 1 & 3 & - 2 \end{matrix}| = (- 2 + 3) \hat{i} - (- 4 + 1) \hat{j} + (6 - 1) \hat{k} = \hat{i} + 3 \hat{j} + 5 \hat{k}$

So, $\overset{鈫�}{A}$ and $\overset{鈫�}{A} 脳 \overset{鈫�}{B}$ are perpendicular vectors.

Hence, these are 2 perpendicular sides of required triangle.

Find third side of triangle

Two sides of triangle are already known.

Third side can be found using vector law of addition.

Third vector is

$\overset{鈫�}{A} + (\overset{鈫�}{A} 脳 \overset{鈫�}{B}) = (2 \hat{i} + \hat{j} - \hat{k}) + (\hat{i} + 3 \hat{j} + 5 \hat{k}) = (2 + 1) \hat{i} + (1 + 3) \hat{j} + (- 1 + 5) \hat{k} = 3 \hat{i} + 4 \hat{j} + 4 \hat{k}$

Conclusion

First side is $2 \hat{i} + \hat{j} - \hat{k}$ .

Second side perpendicular to first one is $\hat{i} + 3 \hat{j} + 5 \hat{k}$

Hypotenuse is $3 \hat{i} + 4 \hat{j} + 4 \hat{k}$ .

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91影视

Find three vectors (none of them parallel to a coordinate axis) which have lengths and directions such that they could be made into a right triangle.

Short Answer

Step by step solution

Required

Results used

Consider two vectors and find a perpendicular vector

Find third side of triangle

Conclusion

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