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:
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:
Show 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.
Updated On: Jun 10, 2026
- \(3\hat i-\hat j+2\hat k\)
- \(3\hat i+\hat j+2\hat k\)
- \(\hat i-3\hat j+4\hat k\)
- \(3\hat i-3\hat j+4\hat k\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
Vectors are quantities that possess both magnitude and direction. When vectors are expressed in terms of the unit vectors
\[
\hat i,\quad \hat j,\quad \hat k,
\]
their addition becomes very straightforward.
If
\[
\vec a=a_1\hat i+a_2\hat j+a_3\hat k
\]
and
\[
\vec b=b_1\hat i+b_2\hat j+b_3\hat k,
\]
then vector addition is performed component-wise according to the rule
\[
\vec a+\vec b
=
(a_1+b_1)\hat i
+
(a_2+b_2)\hat j
+
(a_3+b_3)\hat k.
\]
This means that the coefficients of \(\hat i\), \(\hat j\), and \(\hat k\) are added separately.
Vector addition satisfies the commutative and associative properties and forms one of the fundamental operations in vector algebra.
Step 1: Write the given vectors clearly. We are given \[ \vec a = 2\hat i+\hat j-\hat k \] and \[ \vec b = \hat i-2\hat j+3\hat k. \] To perform vector addition, we compare corresponding components.
Step 2: Add the \(\hat i\)-components. The coefficient of \(\hat i\) in \(\vec a\) is \[ 2. \] The coefficient of \(\hat i\) in \(\vec b\) is \[ 1. \] Therefore, \[ 2+1=3. \] Hence the \(\hat i\)-component of the resultant vector is \[ 3\hat i. \]
Step 3: Add the \(\hat j\)-components. The coefficient of \(\hat j\) in \(\vec a\) is \[ 1. \] The coefficient of \(\hat j\) in \(\vec b\) is \[ -2. \] Therefore, \[ 1+(-2)=-1. \] Hence the \(\hat j\)-component becomes \[ -\hat j. \]
Step 4: Add the \(\hat k\)-components. The coefficient of \(\hat k\) in \(\vec a\) is \[ -1. \] The coefficient of \(\hat k\) in \(\vec b\) is \[ 3. \] Therefore, \[ -1+3=2. \] Hence the \(\hat k\)-component becomes \[ 2\hat k. \]
Step 5: Form the resultant vector. Combining all three components obtained above, \[ \vec a+\vec b = 3\hat i-\hat j+2\hat k. \] This is the required resultant vector.
Step 6: Verification by Component Form. Writing the vectors in ordered form, \[ \vec a=(2,1,-1) \] and \[ \vec b=(1,-2,3). \] Adding corresponding components, \[ (2+1,\;1-2,\;-1+3). \] \[ =(3,-1,2). \] Converting back to vector notation, \[ 3\hat i-\hat j+2\hat k. \] The answer is verified.
Step 7: Final Conclusion. Therefore, \[ \boxed{\vec a+\vec b = 3\hat i-\hat j+2\hat k} \] Hence the correct answer is \[ \boxed{\text{Option (A)}}. \]
Step 1: Write the given vectors clearly. We are given \[ \vec a = 2\hat i+\hat j-\hat k \] and \[ \vec b = \hat i-2\hat j+3\hat k. \] To perform vector addition, we compare corresponding components.
Step 2: Add the \(\hat i\)-components. The coefficient of \(\hat i\) in \(\vec a\) is \[ 2. \] The coefficient of \(\hat i\) in \(\vec b\) is \[ 1. \] Therefore, \[ 2+1=3. \] Hence the \(\hat i\)-component of the resultant vector is \[ 3\hat i. \]
Step 3: Add the \(\hat j\)-components. The coefficient of \(\hat j\) in \(\vec a\) is \[ 1. \] The coefficient of \(\hat j\) in \(\vec b\) is \[ -2. \] Therefore, \[ 1+(-2)=-1. \] Hence the \(\hat j\)-component becomes \[ -\hat j. \]
Step 4: Add the \(\hat k\)-components. The coefficient of \(\hat k\) in \(\vec a\) is \[ -1. \] The coefficient of \(\hat k\) in \(\vec b\) is \[ 3. \] Therefore, \[ -1+3=2. \] Hence the \(\hat k\)-component becomes \[ 2\hat k. \]
Step 5: Form the resultant vector. Combining all three components obtained above, \[ \vec a+\vec b = 3\hat i-\hat j+2\hat k. \] This is the required resultant vector.
Step 6: Verification by Component Form. Writing the vectors in ordered form, \[ \vec a=(2,1,-1) \] and \[ \vec b=(1,-2,3). \] Adding corresponding components, \[ (2+1,\;1-2,\;-1+3). \] \[ =(3,-1,2). \] Converting back to vector notation, \[ 3\hat i-\hat j+2\hat k. \] The answer is verified.
Step 7: Final Conclusion. Therefore, \[ \boxed{\vec a+\vec b = 3\hat i-\hat j+2\hat k} \] Hence the correct answer is \[ \boxed{\text{Option (A)}}. \]
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