Consider three vectors $$A=i^+j^-2k^$$, $$B=i^-j^+k^$$ and $$C=2i^-3j^+4k^$$. A vector X of the form αA $$+$$ βB $$($$α and β are $$numbers)$$ is perpendicular to C. The ratio of α and β is

Question

Consider three vectors $$A=i^+j^-2k^$$, $$B=i^-j^+k^$$ and $$C=2i^-3j^+4k^$$. A vector X of the form αA $$+$$ βB $$($$α and β are $$numbers)$$ is perpendicular to C. The ratio of α and β is

Infinity Learn · Accepted Answer

Vector X of the form αA $$+$$ βB.$$X=$$α$$A+$$β$$B=$$α(i^+j^-2k^)+$$β$$(i^-j^+k^)X=i^($$α$$+$$β$$)+j^($$α$$-$$β$$)+k^(-2$$α$$+$$β$$)A$$ vector X is perpendicular to C, i.e. X · C $$=$$ $$0$$[$$i^($$α$$+$$β$$)+j^($$α$$-$$β$$)+k^(-2$$α$$+$$β$$)]·[$$2i^-3j^+4k^$$]$$=0$$⇒ $$2($$α$$+$$β$$)-3($$α$$-$$β$$)+4(-2$$α$$+$$β$$)=0$$⇒ $$2$$α$$+2$$β$$-3$$α$$+3$$β$$-8$$α$$+4$$β$$=0$$⇒ $$-9$$α$$+9$$β$$=0or$$  α$$=$$β⇒αβ$$=11or$$  α:β$$=1$$:$$1$$

Consider three vectors $A = \hat{i} + \hat{j} - 2 \hat{k}, B = \hat{i} - \hat{j} + \hat{k}$ and $C = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ . A vector X of the form $α$ A + $β$ B ( $α$ and $β$ are numbers) is perpendicular to C. The ratio of $α$ and $β$ is

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Consider three vectors A=i^+j^-2k^, B=i^-j^+k^ and C=2i^-3j^+4k^. A vector X of the form αA + βB (α and β are numbers) is perpendicular to C. The ratio of α and β is

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Consider three vectors $A = \hat{i} + \hat{j} - 2 \hat{k}, B = \hat{i} - \hat{j} + \hat{k}$ and $C = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ . A vector X of the form $α$ A + $β$ B ( $α$ and $β$ are numbers) is perpendicular to C. The ratio of $α$ and $β$ is