Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = -\hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + \hat{j}$ be three vectors. If $(\vec{a} + \lambda\vec{b})$ is perpendicular to $\vec{c}$, then the value of $\lambda$ is

Let $\vec{a} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ and $\vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k}$. Let $\vec{c}$ be a vector such that $\vec{c} \times \vec{a} = \vec{b} \times \vec{c}$ and $\vec{c}\cdot(2\vec{a}-3\vec{b}) = 238\sqrt{2}$ then $|\vec{c}|^2$ is equal to :

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$, $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

If $|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1$, then angle between $\vec{a}$ and $\vec{r}$ is

Match List - I with List - II.

Choose the correct answer from the options given below :

If a unit vector $\hat a$ makes an angle $\dfrac{\pi}{3}$ with $\hat i$, $\dfrac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then the respective value of $\theta$ and $\hat a$ are

Which of the following statements is/are true?

(A) The vector sum of the three sides of a triangle in order is $\vec{0}$

(B) The magnitude $(r)$, direction ratios $(a, b, c)$ and direction cosines $(l, m, n)$ of any vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ are related as $l = \frac{a}{r}, m = \frac{b}{r}, n = \frac{c}{r}$

(C) If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then their cross product is given as $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin \theta$

(D) The cross product of two vectors is commutative

Choose the correct answer from the options given below: