1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?

A
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
B
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
C
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
D
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
2
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$$ and $$\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$$ be three given vectors. If $$\overrightarrow{\mathrm{r}}$$ is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b}=0$$, then $$|\vec{r}|$$ is equal to :

A
$$\frac{11}{7}$$
B
$$\frac{11}{5} \sqrt{2}$$
C
$$\frac{\sqrt{914}}{7}$$
D
$$\frac{11}{7} \sqrt{2}$$
3
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}$ be three vectors. If $\vec{r}$ is a vector such that, $\vec{r} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a}=0$, then $25|\vec{r}|^{2}$ is equal to :
A
336
B
449
C
339
D
560
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let $$\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$$ and $$\vec{b} \cdot \vec{c}=0$$. Consider the following two statements:

(A) $$|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$$ for all $$\lambda \in \mathbb{R}$$.

(B) $$\vec{a}$$ and $$\vec{c}$$ are always parallel.

Then,

A
only (B) is correct
B
both (A) and (B) are correct
C
only (A) is correct
D
neither (A) nor (B) is correct
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