Question

If \[ \vec a=2\hat i+\hat j-\hat k \] and \[ \vec b=\hat i-2\hat j+3\hat k, \] then the value of \[ \vec a+\vec b \] is:

• A. \(3\hat i-\hat j+2\hat k\)
• B. \(3\hat i+\hat j+2\hat k\)
• C. \(\hat i-3\hat j+4\hat k\)
• D. \(3\hat i-3\hat j+4\hat k\)

Hint:

Whenever vectors are expressed in the form \[ a\hat i+b\hat j+c\hat k, \] add or subtract the coefficients of \(\hat i\), \(\hat j\), and \(\hat k\) separately. For example, \[ (2\hat i+3\hat j)+(\hat i-5\hat j) = 3\hat i-2\hat j. \] This component-wise method is the fastest and most reliable approach for vector addition.

Answer 1

The Correct Option is A

Step 1: Write the two vectors.
We have $\vec a=2\hat i+\hat j-\hat k$ and $\vec b=\hat i-2\hat j+3\hat k$. We want $\vec a+\vec b$.

Step 2: Recall how vectors add.
To add vectors, you simply add matching components: the $\hat i$ parts together, the $\hat j$ parts together, and the $\hat k$ parts together.

Step 3: Add the $\hat i$ parts.
The $\hat i$ components are $2$ and $1$. Their sum is $2+1=3$, so we get $3\hat i$.

Step 4: Add the $\hat j$ parts.
The $\hat j$ components are $1$ and $-2$. Their sum is $1+(-2)=-1$, so we get $-\hat j$.

Step 5: Add the $\hat k$ parts.
The $\hat k$ components are $-1$ and $3$. Their sum is $-1+3=2$, so we get $2\hat k$.

Step 6: Combine the results.
Putting the three pieces together gives \[ \vec a+\vec b=3\hat i-\hat j+2\hat k. \] \[ \boxed{3\hat i-\hat j+2\hat k} \]