Let a, b, c be distinct non-negative numbers. If the vectors \( a\hat{i} + a\hat{j} + c\hat{k} \), \( \hat{i} + \hat{k} \) and \( c\hat{i} + c\hat{j} + b\hat{k} \) lie in a plane, then c is
Show Hint
- The Arithmetic Mean of a and b
- The Geometric Mean of a and b
- The Harmonic Mean of a and b
- Equal to zero
The Correct Option is B
Solution and Explanation
If three vectors are coplanar (lie in the same plane), their scalar triple product is zero. The scalar triple product is calculated as the determinant of the matrix formed by the components of the vectors.
Let the vectors be \( \vec{u} = a\hat{i} + a\hat{j} + c\hat{k} \), \( \vec{v} = 1\hat{i} + 0\hat{j} + 1\hat{k} \), and \( \vec{w} = c\hat{i} + c\hat{j} + b\hat{k} \).
For them to be coplanar, \( [\vec{u} \vec{v} \vec{w}] = 0 \). \[ \begin{vmatrix} a & a & c \\ 1 & 0 & 1 \\ c & c & b \end{vmatrix} = 0 \] Expand the determinant along the second row for easier calculation: \\ \[ -1(ab - c^2) + 0 - 1(ac - ac) = 0 \] \[ -ab + c^2 = 0 \] \[ c^2 = ab \] Since a, b, c are non-negative numbers, we can take the square root: \[ c = \sqrt{ab} \]
This is the definition of the Geometric Mean of a and b.
Top Questions on Vectors
Let $\vec{a}$ and $\vec{c}$ be unit vectors such that the angle between them is $\cos^{-1} \left( \frac{1}{4} \right)$. If $\vec{b} = 2\vec{c} + \lambda \vec{a}$. Where $\lambda > 0$ and $|\vec{b}| = 4$, then $\lambda$ is equal to:
- Let \( \mathbf{a} = e\hat{i} + 3\hat{j} + 4\hat{k}, \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k}, \mathbf{c} = 3\hat{i} + \hat{j} + \lambda \hat{k} \) be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of \( \lambda \) is:
If \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \, \mathbf{b} = 2\hat{i} - \hat{j} + 3\hat{k}, \, \mathbf{c} = \hat{i} - 2\hat{j} + \hat{k} \), \(\text{ then a vector of magnitude }\) \( \sqrt{22} \) \(\text{ which is parallel to }\) \( 2\mathbf{a} - \mathbf{b} + \mathbf{c} \) is:
If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $|\vec{a} + \vec{b}| = 1$, then the value of $|\vec{a} \times \vec{b}|$ is:
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} \times \vec{b} = \vec{c}$, $\vec{a} \cdot \vec{c} = 2$ and $\vec{b} \cdot \vec{c} = 1$. If $|\vec{b}| = 1$, then the value of $|\vec{a}|$ is:
Questions Asked in NIMCET exam
- The circle \( x^2 + y^2 + ax + by + \gamma = 0 \) is the image of the circle \( x^2 + y^2 - 6x - 10y + 30 = 0 \) across the line \( 3x + y = 2 \). The value of \( [\alpha + \beta + \gamma] \) is (where \( [.] \) represents the floor function)
- NIMCET - 2025
- Coordinate Geometry
- In a certain code language, 'SUN' is written as 'RWK' and 'MOON' is written as 'LQLK'. Following the same pattern, how will 'STAR' be written?
- NIMCET - 2025
- Coding Decoding
- Which one of the following is NOT a correct statement?
- NIMCET - 2025
- Probability and Statistics
- Let \( A \) and \( B \) be two square matrices of the same order satisfying \( A^2 + 5A + 5I = 0 \) and \( B^2 + 3B + I = 0 \) respectively, where \( I \) is the identity matrix. Then the inverse of the matrix \( C = BA + 2B - 2A + 4I \) is:
- NIMCET - 2025
- Matrix Algebra
- Dynamic RAM (DRAM) stores each bit of data in a separate capacitor. Due to leakage, the stored charge tends to dissipate over time and needs to be refreshed periodically. Consider the following statements: \[\begin{array}{rl} \bullet & \text{P: DRAM requires refreshing because it uses capacitors to store bits.} \\ \bullet & \text{Q: SRAM does not require refreshing because it uses flip-flops instead of capacitors.} \\ \bullet & \text{R: DRAM is faster than SRAM because it needs less frequent access.} \\ \bullet & \text{S: DRAM is more suitable for main memory than SRAM due to its density.} \\ \end{array}\]
- NIMCET - 2025
- Computer Memory