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Q2:
Find the scalar components and magnitude of the vector joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$.
Solution :
Step 1: Establishing the Position Vectors
To determine the vector joining two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in a three-dimensional Cartesian coordinate system, we first define the position vectors $\vec{OP}$ and $\vec{OQ}$ with respect to the origin $O(0, 0, 0)$ [Per the definition of position vectors in $\mathbb{R}^3$]:
$\vec{OP} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$
$\vec{OQ} = x_2\hat{i} + y_2\hat{j} + z_2\hat{k}$
Step 2: Deriving the Vector $\vec{PQ}$
By the Triangle Law of Vector Addition, for the triangle $\triangle OPQ$, the vector joining $P$ to $Q$ is given by the difference of the position vectors [Ref: Vector displacement principle]:
$\vec{PQ} = \vec{OQ} - \vec{OP}$
Substituting the coordinate expressions:
$\vec{PQ} = (x_2\hat{i} + y_2\hat{j} + z_2\hat{k}) - (x_1\hat{i} + y_1\hat{j} + z_1\hat{k})$
$\vec{PQ} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$
Step 3: Identifying Scalar Components
The scalar components of a vector $\vec{A} = a\hat{i} + b\hat{j} + c\hat{k}$ are the coefficients $a, b,$ and $c$. Thus, for vector $\vec{PQ}$:
| Dimension | Scalar Component |
|---|---|
| x-axis | $x_2 - x_1$ |
| y-axis | $y_2 - y_1$ |
| z-axis | $z_2 - z_1$ |
Step 4: Calculating the Magnitude
The magnitude of a vector $\vec{PQ} = a\hat{i} + b\hat{j} + c\hat{k}$ is defined as the Euclidean norm $||\vec{PQ}|| = \sqrt{a^2 + b^2 + c^2}$ [By the Pythagorean Theorem extended to 3D space].
$||\vec{PQ}|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
Final Solution: The scalar components are $(x_2 - x_1), (y_2 - y_1),$ and $(z_2 - z_1)$. The magnitude is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.
More Questions from Class 12 Mathematics Vector Algebra Miscellaneous Exercise on Chapter 10
- Q1: Write down a unit vector in XY-plane, making an angle of $30^{\circ}$ with the positive direction of x-axis.
- Q3: A girl walks 4 km towards west, then she walks 3 km in a direction $30^{\circ}$ east of north and stops. Determine the girl’s displacement from her initial point of departure.
- Q4: If $\vec{a} = \vec{b} + \vec{c}$, then is it true that $|\vec{a}| = |\vec{b}| + |\vec{c}|$? Justify your answer.
- Q5: Find the value of $x$ for which $x(\hat{i} + \hat{j} + \hat{k})$ is a unit vector.
- Q6: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$.
- Q7: If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$, find a unit vector parallel to the vector $2\vec{a} - \vec{b} + 3\vec{c}$.
- Q8: Show that the points A (1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.
- Q9: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $2\vec{a} + \vec{b}$ and $\vec{a} - 3\vec{b}$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
- Q10: The two adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\hat{i} - 2\hat{j} - 3\hat{k}$. Find the unit vector parallel to its diagonal. Also, find its area.
- Q11: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\pm \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$.
- Q12: Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$, $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 15$.
- Q13: The scalar product of the vector $\hat{i} + \hat{j} + \hat{k}$ with a unit vector along the sum of vectors $2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\lambda\hat{i} + 2\hat{j} + 3\hat{k}$ is equal to one. Find the value of $\lambda$.
- Q14: If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$.
- Q15: Prove that $(\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}$.
- Q16: If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a} \cdot \vec{b} \ge 0$ only when
- Q17: Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a} + \vec{b}$ is a unit vector if
- Q18: The value of $\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$ is
- Q19: If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when $\theta$ is equal to
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