Addition & Subtraction of Vectors

Resultant of two vectors $\vec{P}$ and $\vec{Q}$ is of magnitude A. If $\vec{Q}$ is reversed, then the resultant is of magnitude B. The value of $A^2 + B^2$ is

Let $\vec{P} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{Q} = -\hat{i} - \hat{j} - \hat{k}$. The angle between $(\vec{P} - \vec{Q})$ and $\vec{P}$ is

Let $\vec{P} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{Q} = -(\hat{i} + \hat{j} + \hat{k})$. The angle between $(\vec{P} - \vec{Q})$ and $\vec{P}$ is

The resultant $\vec{R}$ of $\vec{P}$ and $\vec{Q}$ is perpendicular to $\vec{P}$. Also $|\vec{P}| = |\vec{R}|$. The angle between $\vec{P}$ and $\vec{Q}$ is ($\tan 45^\circ = 1$)

A vector $\vec{A}$ has magnitude $6$ units is added to vector $\vec{B}$ which is along $x$-axis. The resultant of $\vec{A}$ and $\vec{B}$ is along $y$-axis. If the magnitude of the resultant of $\vec{A}$ and $\vec{B}$ is $3$ times that of $\vec{B}$ then the magnitude of $\vec{B}$ is

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{C}$. If the magnitude of $\vec{B}$ is doubled, the new resultant vector becomes perpendicular to $\vec{A}$. Then the magnitude of $\vec{C}$ is

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{C}$. If the magnitude of $\vec{B}$ is doubled, the new resultant vector becomes perpendicular to $\vec{A}$, then the magnitude of $\vec{C}$ is

The three vector $\vec{A} = 3\hat{i} - 2\hat{j} + \hat{k}, \vec{B} = \hat{i} - 3\hat{j} + 5\hat{k}$ and $\vec{C} = 2\hat{i} - \hat{j} + 4\hat{k}$ will form

The angle between vector $\vec{Q}$ and the resultant of $(2\vec{Q} + 2\vec{P})$ and $(2\vec{Q} - 2\vec{P})$ is:

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at angle $\theta$, then

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is _________${}^\circ$.

If $\vec{a}$ and $\vec{b}$ make an angle $\cos^{-1}\left(\frac{5}{9}\right)$ with each other, then $|\vec{a} + \vec{b}| = \sqrt{2}|\vec{a} - \vec{b}|$ for $|\vec{a}| = n|\vec{b}|$. The integer value of $n$ is _________.

Two forces$\overline{F}_1$ and$\overline{F}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_1$ and $\vec{F}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is _______.

Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is:

Two vectors $\vec{A}$and $\vec{B}$ have equal magnitudes. If magnitude of $\vec{A} + \vec{B}$is equal to two times the magnitude of $\vec{A} - \vec{B}$, then the angle between $\vec{A}$and $\vec{B}$ will be:

Which of the following relations is true for two unit vectors $\hat{A}$ and $\hat{B}$ making an angle $\theta$ to each other?

Match List - I with List - II

Choose the correct answer from the options given below:

Two vectors $\vec{X}$ and $\vec{Y}$ have equal magnitude. The magnitude of ($\vec{X} - \vec{Y}$) is n times the magnitude of ($\vec{X} + \vec{Y}$). The angle between $\vec{X}$ and $\vec{Y}$ is:

Two vectors $\vec{P}$and $\vec{Q}$ have equal magnitudes. If the magnitude of $\vec{P} + \vec{Q}$ is n times the magnitude of $\vec{P} - \vec{Q}$, then angle between $\vec{P}$and $\vec{Q}$ is:

The angle between vector $\left(\vec{A}\right)$ and $\left(\vec{A} - \vec{B}\right)$ is:

Statement I:

Two forces $\left(\vec{P} + \vec{Q}\right)$ and $\left(\vec{P} - \vec{Q}\right)$ where $\vec{P} \perp \vec{Q}$, when act at an angle $\theta$1 to each other, the magnitude of their resultant is $\sqrt{3(P^2 + Q^2)}$, when they act at an angle $\theta$2, the magnitude of their resultant becomes $\sqrt{2(P^2 + Q^2)}$. This is possible only when $\theta_1 < \theta_2$.

Statement II:

In the situation given above.

$\theta$1 = 60${}^\circ$ and $\theta$2 = 90${}^\circ$

In the light of the above statements, choose the most appropriate answer from the options given below :-

Assertion A: If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that , then $\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} = 4\overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{OC}$

Reason R: Polygon law of vector addition yields $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{AD} = 2\overrightarrow{AO}$

In the light of the above statements, choose the most appropriate answer from the options given below:

In an octagon ABCDEFGH of equal side, what is the sum of

$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + \overrightarrow{AF} + \overrightarrow{AG} + \overrightarrow{AH}$,

if, $\overrightarrow{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$

Two vectors of same magnitude have a resultant equal to either of the two vectors. The angle between two vectors is

The resultant of two vector $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{C}$. If the magnitude of $\mathbf{B}$ is doubled, the new resultant vector becomes perpendicular to A. Then, the magnitude of $\mathbf{C}$ is

The sum of two forces $\vec{P}$and $\vec{Q}$is $\vec{R}$ such that $\left|\vec{R}\right| = \left|\vec{P}\right|$. The angle $\theta$ (in degrees) that the resultant of 2$\vec{P}$and $\vec{Q}$ will make with $\vec{Q}$is ..............

The resultant $\mathbf{R}$ of $\mathbf{P}$ and $\mathbf{Q}$ is perpendicular to $\mathbf{P}$. Also $|\mathbf{P}| = |\mathbf{R}|$. The angle between $\mathbf{P}$ and $\mathbf{Q}$ is $\left[\tan 45^\circ = 1\right]$

If $\sqrt{A^2 + B^2}$ represents the magnitude of resultant of two vectors $(\mathbf{A} + \mathbf{B})$ and $(\mathbf{A} - \mathbf{B})$, then the angle between two vectors is

Two vectors $\vec{A}$and $\vec{B}$ have equal magnitudes. The magnitude of $\left(\vec{A} + \vec{B}\right)$ is 'n' times the magnitude of $\left(\vec{A} - \vec{B}\right)$. The angle between $\vec{A}$and $\vec{B}$is -

Two forces P and Q, of magnitude 2F and 3F, respectively, are at an angle $\theta$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle $\theta$ is -

A vector $\vec{A}$is rotated by a small angle $\Delta\theta$ radian ($\Delta\theta$ < <1) to get a new vector $\vec{B}$. In that case is: