Download App

Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors3 Marks
Question
If $\mathrm{u}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{k}, \bar{v}=3 \hat{\mathbf{i}}+\hat{k}$ and $\bar{w}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$ are given vectors, then find

(i) $[\bar{u}+\bar{w}] \cdot[(\bar{w} \times \bar{r}) \times(\bar{r} \times \bar{w})]$

Question is modified.

If $\bar{u}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{k}, \bar{r}=3 \hat{\mathbf{i}}+\hat{k}$ and $\bar{w}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$ are given vectors, then find $[\bar{u}+\bar{w}] \cdot[(\bar{u} \times$

$\bar{r}) \times(\bar{r} \times \bar{w})]$

✓

Answer

$\bar{u}+\bar{w}=(\hat{i}-2 \hat{j}+\hat{k})+(\hat{j}-\hat{k})$

$=\vec{i}-\vec{j}$

$\bar{u} \times \bar{r}=\left|\begin{array}{rrr}\hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 3 & 0 & 1\end{array}\right|$

$\begin{aligned} & =(-2-0) \hat{i}-(1-3) \hat{j}+(0+6) \hat{k} \\ & =-2 \hat{i}+2 \hat{j}+6 \hat{k}\end{aligned}$

$\bar{r} \times \bar{w}=\left|\begin{array}{rrr}\hat{i} & \hat{j} & \hat{k} \\ 3 & 0 & 1 \\ 0 & 1 & -1\end{array}\right|$

$=(0-1) \hat{i}-(-3-0) \hat{j}+(3-0) \hat{k}$

$=-\hat{i}+3 \hat{j}+3 \hat{k}$

$\operatorname{Now}_r(\bar{u}+\bar{w}) \cdot[(\bar{u} \times \bar{r}) \times(\bar{r} \times \bar{w})]=\left|\begin{array}{rrr}1 & -1 & 0 \\ -2 & 2 & 6 \\ -1 & 3 & 3\end{array}\right|$

= 1(6 – 18) + 1 (-6 + 6) + 0
= -12 + 0 + 0 = -12.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Vectors→Vectors — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

$\int\frac{1}{1-\sin\frac{\text{x}}{2}}\text{dx}$
→
Obtain the differential equation by eliminating the arbitrary constants from the following equations:

$y=c_1 e^{2 x}+c_2 e^{5 x}$

→
A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.
→
Using determinants, find the area of the triangle whose vertices are $(1, 4), (2, 3)$ and $(-5, -3)$. Are the given points collinear?
→
Find the adjoint of matrix $A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 3 & 1 & 2 \\ -1 & 1 & 2\end{array}\right]$
→
Find the value of $x$ if the area of $\triangle$ is $35$ square cms with vertices $(x, 4), (2, -6)$ and $(5, 4)$.
→
Show that the area of a triangle ABC, the position vectors of whose vertices are a, b and c

is $\frac{1}{2}[\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}]$

Question is modified.

Show that the area of a triangle $A B C$, the position vectors of whose vertices are $\bar{a}, \bar{b}$ and $\bar{c}$

is $\frac{1}{2}[\bar{a} \times \bar{b}+\bar{b} \times \bar{c}+\bar{c} \times \bar{a}]$.

→
If $x=a\left(t-\frac{1}{t}\right)$ and $y=b\left(t+\frac{1}{t}\right)$, then show that $\frac{d y}{d x}=\frac{b^2 x}{a^2 y}$.
→
Find A (adjoint A) for the matrix $\text{A}=\begin{bmatrix}1 & -2 & 3 \\0 & 2 & -1 \\ -4 & 5 & 2 \end{bmatrix}$.
→
Represent the following families of curves by forming the corresponding differential equation:
$\text{y}=\text{ax}^3$
→