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Let vec(OA) = hat(i)+2hat(j)+2hat(k). In the plane of vec(OA) and hat(i), rotate vec(OA) through 90^(@) about the origin O such that the new position of vec(OA) makes an acute angle with the positive x-axis. The new position of vec(OA)is |
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Answer» `(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))` Let new position of `vec(OA) " is " vec(R)=ahat(i)+bhat(j)+chat(k)` `because vec(OA),vec(r) " and " hat(i) " are coplaner" =|(1,2,2),(a,b,c),(1,0,0)| = 0 IMPLIES b = c` `because vec(r)_|_vec(OA) implies a+2b+2c=0impliesa=-4b{because b=c}` `therefore vec(r)=-4bhat(j)+bhat(j)+bhat(k)=-b(4hat(i)-hat(j)-hat(k))` Also `|vec(r)|=|vec(OA)|impliesb=pm(1)/(sqrt(2))impliesvec(r)=pm(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))` `because vec(r)` makes acute angle withpositive x-axis `implies vec(r)=(1)/(sqrt(2))(4hat(i)-hat(j)-hat(k))` |
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