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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
Solution :
Given Variables & Initial Setup
Let $\vec{a}$ and $\vec{b}$ be two unit vectors. By definition of unit vectors:
- $|\vec{a}| = 1$ [Definition of unit vector]
- $|\vec{b}| = 1$ [Definition of unit vector]
- The angle between $\vec{a}$ and $\vec{b}$ is $\theta$. Thus, $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta = \cos \theta$ [By the definition of dot product].
Step 1: Establishing the Condition for the Resultant
We are given that $\vec{c} = \vec{a} + \vec{b}$ is also a unit vector. Therefore:
$|\vec{a} + \vec{b}| = 1$
Squaring both sides to eliminate the square root typically found in the magnitude formula:
$|\vec{a} + \vec{b}|^2 = 1^2$
$(\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = 1$ [By the property $|\vec{v}|^2 = \vec{v} \cdot \vec{v}$]
Step 2: Algebraic Expansion and Substitution
Expanding the dot product using the distributive property:
$\vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{b} = 1$
$|\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 = 1$ [Since $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ by commutativity of dot product]
Substitute the known magnitudes ($|\vec{a}| = 1, |\vec{b}| = 1$) and the dot product ($\vec{a} \cdot \vec{b} = \cos \theta$):
$(1)^2 + 2 \cos \theta + (1)^2 = 1$
Step 3: Solving for $\theta$
$1 + 2 \cos \theta + 1 = 1$
$2 + 2 \cos \theta = 1$
$2 \cos \theta = 1 - 2$
$2 \cos \theta = -1$
$\cos \theta = -\frac{1}{2}$
Since $\theta$ is the angle between two vectors, $0 \le \theta \le \pi$. The value of $\theta$ for which $\cos \theta = -\frac{1}{2}$ is:
$\theta = \frac{2\pi}{3}$ or $120^\circ$
Geometric Representation
Final Solution: $\theta = \frac{2\pi}{3}$
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.
- Q2: Find the scalar components and magnitude of the vector joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$.
- 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
- 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
CBSE Solutions for Class 12 Mathematics Vector Algebra
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