Download App

Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product5 Marks
Question
Let $\vec{\text{u}},\vec{\text{v}}$ and $\vec{\text{w}}$ be vectors such $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=\vec{0}.$ If $|\vec{\text{u}}|=3,|\vec{\text{v}}|=4$ and $|\vec{\text{w}}|=5,$ then find $\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}.$
✓

Answer

Here, $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=0$
Squaring both the sides,
$\big(\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}\big)^2=(0)^2$
$|\vec{\text{u}}|^2+|\vec{\text{v}}|^2+|\vec{\text{w}}|^2+2\vec{\text{u}}\vec{\text{v}}+2\vec{\text{v}}\vec{\text{w}}+2\vec{\text{w}}\vec{\text{u}}=0$
$(3)^2+(4)^2+(5)^2+2\big(\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}\big)=0$
$9+16+25+2\big(\vec{\text{u}}\vec{\text{v}}+\vec{\text{v}}\vec{\text{w}}+\vec{\text{w}}\vec{\text{u}}\big)=0$
$2\big(\vec{\text{u}}\vec{\text{v}}+\vec{\text{v}}\vec{\text{w}}+\vec{\text{w}}\vec{\text{u}}\big)=-50$
$\vec{\text{u}}\vec{\text{v}}+\vec{\text{v}}\vec{\text{w}}+\vec{\text{w}}\vec{\text{u}}=\frac{-50}{2}$
$\vec{\text{u}}\vec{\text{v}}+\vec{\text{v}}\vec{\text{w}}+\vec{\text{w}}\vec{\text{u}}=-25$

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 "5 Marks Questions" from Scalar Or Dot Product→Scalar Or Dot Product — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the matrix $X$ satisfying the equation:
$\begin{bmatrix}2 & 1 \\5 & 3 \end{bmatrix}\text{X}\begin{bmatrix}5 & 3 \\3 & 2 \end{bmatrix}=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}.$
→
Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).
→
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is $\frac{4r}{3}$.
→
Find the equation of the curve passing through the point $\Big(0\frac{\pi}{4}\Big)$ whose differential equation is $\sin\text{x}\ \cos\text{y}\ \text{dx}+\cos\text{x}\ \sin\text{y}\ \text{dy}=0.$
→
By using the properties of definite integrals, evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$
→
Sketch the graph y = |x + 3|. Evaluate $\int\limits_{-6}^{0}|\text{x}-3|\text{dx} $ . What does this value of the integral represent on the graph.
→
Solve the following differential equation
$\sin\Big(\frac{\text{dy}}{\text{dx}}\Big)=\text{K};\text{y}(0)=1$
→
If $\text{A}=\begin{bmatrix}4 & 3 \\ 2 & 5 \end{bmatrix},$ find x and y such that $A^2 = zA + yI = 0.$ Hence, evaluate $A^{-1}.$
→
Prove that $x^2 - y^2 = c(x^2 + y^2)^2$ is the general solution of differential equation $(x^3-3xy^2)dx=(y^3-3x^2y)dy$, where c is a parameter.
→
Prove that the line $\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)+\lambda\big(3\hat{\text{i}}-\hat{\text{j}}\big)$ and $\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{k}}\big)+\mu\big(2\hat{\text{i}}+3\hat{\text{k}}\big)$ intersect and find their point of intersection.
→