Vector Algebra · Mathematics · NDA

If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then what is the value of $l^{2} + m^{2} - n^{2}$?

Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \times \vec{a}) = \alpha \hat{i} - \beta \hat{j} + \gamma \hat{k}$, then what is the value of $\alpha + \beta + \gamma$?

If a vector of magnitude 2 units makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?

Consider the following in respect of moment of a force:

1. The moment of force about a point is independent of point of application of force.

2. The moment of a force about a line is a vector quantity.

Which of the statements given above is/are correct?

For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdot\hat{j}\right)\left(\vec{r}\times\hat{j}\right) + \left(\vec{r}\cdot\hat{k}\right)\left(\vec{r}\times\hat{k}\right)$ equal to?

Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\vec{a}$ and $\vec{a} - \vec{b}$?

Consider the following in respect of the vectors $\rm \vec{a}=(0,1,1)$ and $\rm \vec{b}=(1,0,1) $ :

1. The number of unit vectors perpendicular to both $\rm \vec{a}$ and $\rm \vec{b}$ is only one.

2. The angle between the vectors is $\frac{\pi}{3}$.

Which of the statements given above is/are correct?

Consider the following points :

1. (-1, -3, 1)

2. (-1, 3, 2)

3. (-2, 5, 3)

Which of the above points lie on the line joining A and B ?  

Consider the following statements :

1. Dot product over vector addition is distributive

2. Cross product over vector addition is distributive

3. Cross product of vectors is associative

Which of the above statements is/are correct ?

Let  $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that  $\vec{a}\times \vec{b} = \vec{c} $. Consider the following statements:

1. $\vec a$ is unique if $\vec b$ and $\vec c$ are given

2. $\vec c$ is unique if $\vec a$ and $\vec b$ are given

Which of the above statements is/are correct?

A vector $\vec r=a \hat i+b \hat j$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?

Consider the following statements in respect of a vector $\vec c=\vec a+\vec b$, where $|\vec a|=|\vec b|\ne0$:

1. $\vec c$ is perpendicular to $(\vec a-\vec b).$

2. $\vec c$ is perpendicular to $\vec a \times \vec b.$

Which of the above statement is/are correct?

Consider the following statements:

1. The cross product of two unit vectors is always a unit vector.

2. The dot product of two unit vectors is always unity.

3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.

Which of the above statements are not correct?