If the resultant of three vectors A = -i + 2j + 3k, B = -2i - j - 4k and C is a vector in the positive z-direction with a magnitude of 2 units, then the vector C =

The correct answer is Option 1. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Physics Q84 – AP-EAMCET 2026

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Home Physics Vector AlgebraQuestion #580

If the resultant of three vectors

\vec{A} = - \hat{i} + 2 \hat{j} + 3 \hat{k}

\vec{B} = - 2 \hat{i} - \hat{j} - 4 \hat{k}

and

\vec{C}

is a vector in the positive

z

-direction with a magnitude of 2 units, then the vector

\vec{C} =

Clue 1 of 3: Vector Addition and Resultant

The resultant of multiple vectors is their vector sum. If $\vec{R}$ is the resultant of $\vec{A}$ , $\vec{B}$ , and $\vec{C}$ , then $\vec{R} = \vec{A} + \vec{B} + \vec{C}$ .

🧠 Full Solution Path

Step 1: Define the resultant vector✦ Active

The problem states that the resultant of the three vectors is a vector in the positive $z$ -direction with a magnitude of 2 units. Therefore, the resultant vector $\vec{R}$ can be written as:

\vec{R} = 2 \hat{k}

💡 Teacher's Secret Hint

Remember that $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ are unit vectors along the positive $x$ , $y$ , and $z$ axes, respectively.

Step 2: Express the unknown vector

\vec{C}

in terms of the resultant and known vectors○ Expand

The resultant vector $\vec{R}$ is the sum of the three vectors $\vec{A}$ , $\vec{B}$ , and $\vec{C}$ . So, $\vec{R} = \vec{A} + \vec{B} + \vec{C}$ . We need to find $\vec{C}$ , so we can rearrange the equation:

\vec{C} = \vec{R} - (\vec{A} + \vec{B})

💡 Teacher's Secret Hint

Vector subtraction is performed component-wise, similar to addition. Be careful with the signs when distributing the negative sign.

Step 3: Calculate the sum of vectors

\vec{A}

and

\vec{B}

○ Expand

Given $\vec{A} = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $\vec{B} = - 2 \hat{i} - \hat{j} - 4 \hat{k}$ . Add the corresponding components:

\vec{A} + \vec{B} = (- 1 - 2) \hat{i} + (2 - 1) \hat{j} + (3 - 4) \hat{k}

\vec{A} + \vec{B} = - 3 \hat{i} + \hat{j} - \hat{k}

💡 Teacher's Secret Hint

Always group the $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ components separately to avoid errors in addition or subtraction.

Step 4: Calculate vector

\vec{C}

○ Expand

Substitute the values of $\vec{R}$ from Step 1 and $(\vec{A} + \vec{B})$ from Step 3 into the equation from Step 2:

\vec{C} = 2 \hat{k} - (- 3 \hat{i} + \hat{j} - \hat{k})

Distribute the negative sign and combine like components:

\vec{C} = 2 \hat{k} + 3 \hat{i} - \hat{j} + \hat{k}

\vec{C} = 3 \hat{i} - \hat{j} + (2 + 1) \hat{k}

\vec{C} = 3 \hat{i} - \hat{j} + 3 \hat{k}

💡 Teacher's Secret Hint

Double-check your signs, especially when subtracting a vector. A common mistake is to forget to change the sign of all components within the parentheses.

Step 5: Compare the result with the given options○ Expand

The calculated vector $\vec{C} = 3 \hat{i} - \hat{j} + 3 \hat{k}$ matches option 1.

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Physics Question 84 – AP-EAMCET 2026

Clue 1 of 3: Vector Addition and Resultant

🧠 Full Solution Path