Question:
If \(\overset{̄}{a} = \hat{i}+\hat{j}+\hat{k},\overset{̄}{b} = \hat{i}-\hat{j}+2\hat{k}\) and \(\overset{̄}{c} = x\hat{i}+(x-2)\hat{j}-\hat{k}\) are three vectors in which \(\overset{̄}{c}\) lies in the plane of \(\overset{̄}{a}\) and \(\overset{̄}{b}\), then \(x = \cdots\)
If \(\overset{̄}{a} = \hat{i}+\hat{j}+\hat{k},\overset{̄}{b} = \hat{i}-\hat{j}+2\hat{k}\) and \(\overset{̄}{c} = x\hat{i}+(x-2)\hat{j}-\hat{k}\) are three vectors in which \(\overset{̄}{c}\) lies in the plane of \(\overset{̄}{a}\) and \(\overset{̄}{b}\), then \(x = \cdots\)
Show Hint
Three vectors are coplanar when their scalar triple product is zero.
Updated On: Oct 1, 2026
- \(0\)
- \(1\)
- \(-4\)
- \(-2\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Coplanarity condition:
\(\vec c\) lies in the plane of \(\vec a\) and \(\vec b\) when \([\vec a\ \vec b\ \vec c]=0\).
Step 2: Form the determinant:
\[ \begin{vmatrix}1&1&1\\1&-1&2\\x&x-2&-1\end{vmatrix}=0 \]
Step 3: Expand:
\(1[(-1)(-1)-2(x-2)]-1[(1)(-1)-2x]+1[(x-2)-(-1)x]\)
\(=(1-2x+4)-(-1-2x)+(2x-2)=5-2x+1+2x+2x-2=4+2x\).
Step 4: Solve:
\(4+2x=0\) gives \(x=-2\). Option D.
Final Answer:
x = -2. \[ \boxed{\text{(D) }-2} \]
\(\vec c\) lies in the plane of \(\vec a\) and \(\vec b\) when \([\vec a\ \vec b\ \vec c]=0\).
Step 2: Form the determinant:
\[ \begin{vmatrix}1&1&1\\1&-1&2\\x&x-2&-1\end{vmatrix}=0 \]
Step 3: Expand:
\(1[(-1)(-1)-2(x-2)]-1[(1)(-1)-2x]+1[(x-2)-(-1)x]\)
\(=(1-2x+4)-(-1-2x)+(2x-2)=5-2x+1+2x+2x-2=4+2x\).
Step 4: Solve:
\(4+2x=0\) gives \(x=-2\). Option D.
Final Answer:
x = -2. \[ \boxed{\text{(D) }-2} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Vector Algebra Questions
- If vector $\vec{r}$ with d.c.s. $l, m, n$ is equally inclined to the co-ordinate axes, then the total number of such vectors is
If a, b, c are position vectors of points A, B, C respectively, with 2a + 3b -5c = 0 , then the ratio in which point C divides segment AB is
If a and b are two vectors such that I\(\vec {a}\)I + I\(\vec {b}\)I = \(\sqrt 2\) with \(\vec {a}\).\(\vec {b}\) = –1, then the angle between \(\vec {a}\) and \(\vec {b}\) is
- The vector projection of b on a , where a = 3\(\hat {i}\) + 2\(\hat {j}\) + 5\(\hat {k}\) and b = 7\(\hat {i}\) - 5\(\hat {j}\) - \(\hat {k}\) is
The ratio in which the plane r.(\(\hat i\) -2\(\hat j\) + 3\(\hat k\) ) =17 divides the line joining the points -2\(\hat i\)+4\(\hat j\)+7\(\hat k\) and 3\(\hat i\)-5\(\hat j\)+8\(\hat k\) is
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions