Download App

Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
If a vector$​​\vec{\text{a}}$ is perpendicular to two non-collinear vectors $\vec{\text{b}}$ and $\vec{\text{c}},$ then show that $​​\vec{\text{a}}$ is perpendicular to every vector in the plane of $\vec{\text{b}}$ and $\vec{\text{c}}.$
✓

Answer

Given that $\vec{\text{a}}$ is perpendicular to $\vec{\text{b}}$ and $\vec{\text{c}}.$
$\Rightarrow\vec{\text{a}}.\vec{\text{b}}=0$ and $\vec{\text{a}}.\vec{\text{c}}=0\dots(1)$
Now, let $\vec{\text{r}}$ be any vector in the plane of $\vec{\text{b}}$ and $\vec{\text{c}}.$
Then, $\vec{\text{r}}$ is the linear combination of $\vec{\text{b}}$ and $\vec{\text{c}}.$
$\vec{\text{r}}=\text{x}\vec{\text{b}}+\text{y}\vec{\text{c}}, $ for some x and y.
Now,
$​​\vec{\text{a}}.​​\vec{\text{r}}$
$=​​\vec{\text{a}}.\big(\text{x}\vec{\text{b}}+\text{y}\vec{\text{c}}\big)$
$=\text{x}\big(\vec{\text{a}}.\vec{\text{b}}\big)+\text{y}\big(\vec{\text{a}}.\vec{\text{c}}\big)$
$=\text{x}(0)+\text{y}(0)$ [From(1)]
$=0$
Thus, $\vec{\text{a}}$ is perpendicular to $\vec{\text{r}}.$
That is, $\vec{\text{a}}$ is perpendicular to every vector in the plane of $\vec{\text{b}}$ and $\vec{\text{c}}.$

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 "3 Marks Question" 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

Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Show that '*' is both commutative and associative.
→
Prove that y= $\frac{4\sin\theta}{(2+\cos \theta)}-\theta$ is an increasing function of $\theta \text{ in }\bigg(0,\frac{\pi}{2}\bigg).$
→
If $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then show that the vectores $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}$ are orthonal.
→
Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
→
Write a value of $\int\frac{\sin\text{x}-\cos\text{x}}{\sqrt{1+\sin2\text{x}}}\text{ dx}$
→
Prove the following results:
$\frac{9\pi}{4}-\frac{9}{4}\sin^{-1}\frac{1}{3}=\frac{9}{4}\sin^{-1}\frac{2\sqrt2}{3}$
→
Integrate the following integrals:
$\int\cos3\text{x}\cos4\text{x dx}$
→
If $\text{A}=\begin{bmatrix}-4 & -3 & -3 \\1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix},$ show that adj A = A.
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+1}{\text{x}^2-1}\ \text{dx}$
→
Find the value of $\int \frac{1}{\sqrt{\sin ^3 x \sin (x+\alpha)}} d x$.
→