If $a$,$b$,$c$ are the $p^{th}$,$q^{th}$,$r^{th}$ terms of an $A.P.$ and $\vec x = (q - r)\hat i + (r - p)\hat j + (p - q)\hat k$ and $\vec y = a\hat i + b\hat j + c\hat k$,then:
- A$\vec x, \vec y$ are parallel vectors
- B$\vec x \times \vec y = \hat i + \hat j + \hat k$
- C$\vec x \cdot \vec y = 1$
- D$\vec x, \vec y$ are orthogonal vectors
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If $a, b, c$ are the position vectors of the points $A, B, C$ respectively,then match the items of List-$I$ with those of List-$II$.
List-$I$ List-$II$ $A$. $a = 2\hat{i} + 3\hat{j} + 4\hat{k}, b = 3\hat{i} + 4\hat{j} + 2\hat{k}, c = 4\hat{i} + 2\hat{j} + 3\hat{k}$ $I$. $\triangle ABC$ is an equilateral triangle $B$. $a = \hat{i} + 2\hat{j} + 3\hat{k}, b = 3\hat{i} + 4\hat{j} + 7\hat{k}, c = -3\hat{i} - 2\hat{j} - 5\hat{k}$ $II$. $\triangle ABC$ is an isosceles triangle $C$. $a = 2\hat{i} - \hat{j} + \hat{k}, b = \hat{i} - 3\hat{j} - 5\hat{k}, c = -3\hat{i} - 4\hat{j} - 4\hat{k}$ $III$. $\triangle ABC$ is a right-angled triangle $D$. $a = \hat{i} + \hat{j} + \hat{k}, b = \hat{i} + 2\hat{j} + 3\hat{k}, c = 2\hat{i} - \hat{j} + \hat{k}$ $IV$. $A, B, C$ are collinear
The correct match is:
| List-$I$ | List-$II$ |
| $A$. $a = 2\hat{i} + 3\hat{j} + 4\hat{k}, b = 3\hat{i} + 4\hat{j} + 2\hat{k}, c = 4\hat{i} + 2\hat{j} + 3\hat{k}$ | $I$. $\triangle ABC$ is an equilateral triangle |
| $B$. $a = \hat{i} + 2\hat{j} + 3\hat{k}, b = 3\hat{i} + 4\hat{j} + 7\hat{k}, c = -3\hat{i} - 2\hat{j} - 5\hat{k}$ | $II$. $\triangle ABC$ is an isosceles triangle |
| $C$. $a = 2\hat{i} - \hat{j} + \hat{k}, b = \hat{i} - 3\hat{j} - 5\hat{k}, c = -3\hat{i} - 4\hat{j} - 4\hat{k}$ | $III$. $\triangle ABC$ is a right-angled triangle |
| $D$. $a = \hat{i} + \hat{j} + \hat{k}, b = \hat{i} + 2\hat{j} + 3\hat{k}, c = 2\hat{i} - \hat{j} + \hat{k}$ | $IV$. $A, B, C$ are collinear |
The correct match is:
If the position vectors of $P$ and $Q$ are $\hat{i}+2 \hat{j}-7 \hat{k}$ and $5 \hat{i}-3 \hat{j}+4 \hat{k}$ respectively,then the cosine of the angle between $\overrightarrow{PQ}$ and $z$-axis is
If $a$ and $b$ are unit vectors and $\alpha$ is the angle between them,then $a+b$ is a unit vector when $\cos \alpha=$
Let $\vec{\alpha}=4 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$,then the value of $5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})$ is
DifficultJEE MAIN 2023
View SolutionIf the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $-\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is $1,$ then $\lambda$ is equal to ..... .
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