Vector Algebra Problem Analysis
We are given the vectors:
- $\vec{a} = \hat{i} + \hat{j} + \hat{k}$
- $\vec{b} = \hat{j} - \hat{k}$
And the following conditions relating them to an unknown vector $\vec{c}$:
- The cross product: $\vec{a} \times \vec{c} = \vec{b}$
- The dot product: $\vec{a} \cdot \vec{c} = 3$
The objective is to determine the value of the expression $\vec{c} \cdot (\vec{a} - 2\vec{b})$.
Solving the Vector Expression
First, expand the target expression using the properties of the dot product:
$ \vec{c} \cdot (\vec{a} - 2\vec{b}) = \vec{c} \cdot \vec{a} - \vec{c} \cdot (2\vec{b}) $
Rearrange and use the commutative property of the dot product ($\vec{c} \cdot \vec{a} = \vec{a} \cdot \vec{c}$):
$ = \vec{a} \cdot \vec{c} - 2(\vec{b} \cdot \vec{c}) $
Calculating Required Dot Products
1. Value of $\vec{a} \cdot \vec{c}$:
The problem explicitly states that $\vec{a} \cdot \vec{c} = 3$. No calculation is needed.
2. Value of $\vec{b} \cdot \vec{c}$:
We use the given cross product condition $\vec{a} \times \vec{c} = \vec{b}$.
To find $\vec{b} \cdot \vec{c}$, we can take the dot product of both sides of the cross product equation with $\vec{c}$:
$ \vec{c} \cdot (\vec{a} \times \vec{c}) = \vec{c} \cdot \vec{b} $
The left side, $\vec{c} \cdot (\vec{a} \times \vec{c})$, is a scalar triple product, specifically $[\vec{c}, \vec{a}, \vec{c}]$.
A key property of the scalar triple product is that it equals zero if any two of the vectors involved are identical. Here, the first and third vectors are both $\vec{c}$.
Therefore, $\vec{c} \cdot (\vec{a} \times \vec{c}) = 0$.
This leads to $\vec{c} \cdot \vec{b} = 0$, which means $\vec{b} \cdot \vec{c} = 0$.
Final Calculation
Now, substitute the values of the dot products into the simplified expression:
$ \vec{c} \cdot (\vec{a} - 2\vec{b}) = \vec{a} \cdot \vec{c} - 2(\vec{b} \cdot \vec{c}) $
$ = 3 - 2(0) $
$ = 3 - 0 $
$ = 3 $
The final value of the expression is 3.
Similar Questions
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
- Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is_________.
- For some $\theta \in \left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $l_2$, respectively. If $e_1^2 = e_2^2(\sec^2\theta+1)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right)\tan^2\theta$ is equal to ________
- If foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{12} = 1$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ coincide, then the eccentricity of the hyperbola, with foci $(0, \pm \sqrt{10})$ passing through the point $(a, a + 1)$, is
- Let the curves $y^2 = kx$ and $xy = -1$ have a common tangent, whose slope is $\frac{1}{2}$. Then $k$ can NOT lie in the interval
Important Questions from Coordinate Geometry
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
