If the vectors a -2 i - y - z j -5 k - and b - x - y i -3 j - z - x k - are equal x-y- z -

Given: nbsp;a #8594;=2i^ (y+z)j^+5k^ #160;and #160;b #8594;=(x+y)i^+3j^+(z+x)k^ nbsp;are equal Two vectors are equal if and only if their correspo ...

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Standard XII

Mathematics

Equal Vectors

If the vector...

Question

If the vectors $\vec{a} = 2 \hat{i} - (y + z) \hat{j} + 5 \hat{k} and \vec{b} = (x + y) \hat{i} + 3 \hat{j} + (z + x) \hat{k}$ are equal x+y+z= _________________.

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Solution

Given: $\vec{a} = 2 \hat{i} - (y + z) \hat{j} + 5 \hat{k} and \vec{b} = (x + y) \hat{i} + 3 \hat{j} + (z + x) \hat{k}$ are equal

Two vectors are equal if and only if their corresponding components are equal.
Thus,

$2 = x + y, - (y + z) = 3, 5 = z + x Now, x + y = 2 . . (1) - y - z = 3 . . (2) z + x = 5 . . (3) Subtracting (1) from (3), we get z - y = 3 . . (4) Adding (2) and (4), we get - 2 y = 6 \Rightarrow y = - 3 Substituting the value of y in (1) and (4), we get x - 3 = 2 \Rightarrow x = 5 and z + 3 = 3 \Rightarrow z = 0 Thus, x + y + z = 5 - 3 + 0 = 2$

Hence, x + y + z = 2.

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If the vectors a→=2i^-(y+z)j^+5k^ and b→=(x+y)i^+3j^+(z+x)k^ are equal x+y+z= _________________.

If the vectors $\vec{a} = 2 \hat{i} - (y + z) \hat{j} + 5 \hat{k} and \vec{b} = (x + y) \hat{i} + 3 \hat{j} + (z + x) \hat{k}$ are equal x+y+z= _________________.