Vector Algebra — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
Let $\vec{a},\vec{b},\vec{c},\vec{a}+\vec{b},\vec{b}+\vec{c},\vec{a}+\vec{b}+\vec{c}$ be unit vectors.

What is the angle between $\vec{a}$ and $\vec{b}$ ?

Directions : For the next two (02) items that follow :
Let $\vec{a},\vec{b},\vec{c},\vec{a}+\vec{b},\vec{b}+\vec{c},\vec{a}+\vec{b}+\vec{c}$ be unit vectors.

What is the angle between $\vec{a}$ and $\vec{c}$ ?

Directions : For the next two (02) items that follow :
Let $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{b}\times \vec{c}=\vec{a}$

Consider the following statements :
I. $\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs.
II. $\vec{a},\vec{b},\vec{c}$ are unit vectors.
Which of the statements given above is/are correct ?

Directions : For the next two (02) items that follow :
Let $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{b}\times \vec{c}=\vec{a}$

Consider the following statements :
I. $(\vec{a}\times \vec{b})\cdot \vec{c}+(\vec{b}\times \vec{c})\cdot \vec{a}=(\vec{c}\times \vec{a})\cdot \vec{b}$
II. $\{(\vec{a}\times \vec{b})\times (\vec{b}\times \vec{c})\}\cdot \vec{b}=1$
Which of the statements given above is/are correct ?

Directions : For the next two (02) items that follow :
Let $\vec{a},\vec{b},\vec{c}$ be unit vectors. Further, $\vec{a}$ is perpendicular to $\vec{b}$; $\vec{c}$ makes an angle $\frac{\pi }{3}$ with both $\vec{a}$ and $\vec{b}$; and $\vec{c}=p\vec{a}+q\vec{b}+r(\vec{a}\times \vec{b})$.

What is the value of $(p+q)$ ?

Directions : For the next two (02) items that follow :
Let $\vec{a},\vec{b},\vec{c}$ be unit vectors. Further, $\vec{a}$ is perpendicular to $\vec{b}$; $\vec{c}$ makes an angle $\frac{\pi }{3}$ with both $\vec{a}$ and $\vec{b}$; and $\vec{c}=p\vec{a}+q\vec{b}+r(\vec{a}\times \vec{b})$.

What is the value of $r^2$ ?

A line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes. If $\vec{a}=({\sin }^2\alpha )\hat{i}+({\sin }^2\beta )\hat{j}+({\sin }^2\gamma )\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}+\hat{k}$, then what is $\vec{a}\cdot \vec{b}$ equal to?

Consider the following statements in respect of a vector $\vec{d}=(\vec{a}\times \vec{b})\times \vec{c}$ :
I. $\vec{d}$ is coplanar with $\vec{a}$ and $\vec{b}$.
II. $\vec{d}$ is perpendicular to $\vec{c}$.
Which of the statements given above is/are correct?

The position vectors of three points $A$, $B$ and $C$ are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively such that $3\vec{a}-4\vec{b}+\vec{c}=\vec{0}$. What is $AB:BC$ equal to?

The position vectors of three points $A$, $B$ and $C$ are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively, where $\vec{c}=({\cos }^2\theta )\vec{a}+({\sin }^2\theta )\vec{b}$. What is $(\vec{a}\times \vec{b})+(\vec{b}\times \vec{c})+(\vec{c}\times \vec{a})$ equal to?