If veca⋅ hat i= veca ⋅( hat i+ hat j)= veca⋅ ( hat i+ hat j+ hat k)=1 then veca=

Question

If veca⋅ hat i= veca ⋅( hat i+ hat j)= veca⋅ ( hat i+ hat j+ hat k)=1 then veca= (A)  hat i+ hat j (B)  hat i- hat k (C)  hat i (D)  hat i+ hat j- hatk

Tardigrade Admin · Accepted Answer

Let veca =a1 hati + a2 hatj + a3 hatk Given, veca . hati = veca . ( hati + hatj) = veca .( hati + hatj + hatk) = 1 ∴ a1 =a1 + a2 =a1+a2 + a3 = 1 ⇒ a1 = 1 , a2 = 0 , a3 = 0 ∴ veca = hati

Q. If $a \cdot \hat{i} = a \cdot (\hat{i} + \hat{j}) = a \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$ then $a$ =

$\hat{i} + \hat{j}$

$\hat{i} - \hat{k}$

$\hat{i}$

$\hat{i} + \hat{j} - \hat{k}$

Let $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$
Given, $a . \hat{i} = a . (\hat{i} + \hat{j}) = a . (\hat{i} + \hat{j} + \hat{k}) = 1$
$∴ a_{1} = a_{1} + a_{2} = a_{1} + a_{2} + a_{3} = 1$
$\Rightarrow a_{1} = 1, a_{2} = 0, a_{3} = 0$
$∴ a = \hat{i}$

Q. If a⋅i^=a⋅(i^+j^​)=a⋅(i^+j^​+k^)=1 then a=

i^+j^​

i^−k^

i^

i^+j^​−k^