You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Let a=i+2j+...

Question

Let $a = i + 2 j + k, b = i - j + k$ and $c = i + j - k$ . A vector in the plane of $a$ and $b$ , whose projection on $c$ is $\frac{1}{\sqrt{3}}$ is

- 4 i + j - 4 k

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

3 i + j - 3 k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

i + j - 2 k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

4 i + j - 4 k

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $- 4 i + j - 4 k$
A vector $d$ in the plane of $a$ and $b$ is given by, $d = λ a + μ b = (λ + μ) i + (2 λ - μ) j + (λ + μ) k,$
Now the projection of $d$ on $c$ is $\frac{c \cdot d}{| c |} = \frac{λ + μ + (2 λ - μ) - (λ + μ)}{\sqrt{3}} = \frac{1}{\sqrt{3}}$ (given)
$\Rightarrow 2 λ - μ = 1$ . Thus $d = (3 λ - 1) i + j + (3 λ - 1) k,$
So for $λ = - 1$ , $d = - 4 i + j - 4 k$

Suggest Corrections

Similar questions

Let $\to a = i + 2 j + k, \to b = i - j + k a n d \to c = i + j - k,$ a vector in the plane $\to a a n d \to b$ whose projection on $\to c i s \frac{1}{\sqrt{3}} i s . . . . . . . .$

Q. Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
(i)

\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}

(ii)

\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}

(iii)

\vec{a} = 11 \hat{i}, \vec{b} = 2 \hat{j}, \vec{c} = 13 \hat{k}

(iv)

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}

Q. Find the angle between the vectors

\vec{a} and \vec{b}

, where
(i)

\vec{a} = \hat{i} - \hat{j} and \vec{b} = \hat{j} + \hat{k}

(ii)

\vec{a} = 3 \hat{i} - 2 \hat{j} - 6 \hat{k} and \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}

(iii)

\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} and \vec{b} = 4 \hat{i} + 4 \hat{j} - 2 \hat{k}

(iv)

\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} and \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}

(v)

\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}

Q. If

a = 3 i + 3 j + 4 k, b = i + j + k

and

c = 4 i + 2 j + 3 k

then

\frac{1}{5 \sqrt{2}} | a \times (b \times c) |

is equal to

Q. Find

[\vec{a} \vec{b} \vec{c}]

, when
(i)

\vec{a} = 2 \hat{i} - 3 \hat{j}, \vec{b} = \hat{i} + \hat{j} - \hat{k} and \vec{c} = 3 \hat{i} - \hat{k}

(ii)

\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k} and \vec{c} = \hat{j} + \hat{k}

(iii)

\vec{a} = 2 \hat{i} + 3 \hat{j} + \hat{k}, \vec{b} = \hat{i} - 2 \hat{j} + \hat{k} and \vec{c} = - 3 \hat{i} + \hat{j} + 2 \hat{k}

Join BYJU'S Learning Program

Let a=i+2j+k, b=i−j+k and c=i+j−k. A vector in the plane of a and b, whose projection on c is 1√3 is

The correct option is A −4i+j−4kA vector d in the plane of a and b is given by, d=λa+μb=(λ+μ)i+(2λ−μ)j+(λ+μ)k, Now the projection of don c is c⋅d|c|=λ+μ+(2λ−μ)−(λ+μ)√3=1√3(given) ⇒2λ−μ=1. Thus d=(3λ−1)i+j+(3λ−1)k,So for λ=−1, d=−4i+j−4k

Let $a = i + 2 j + k, b = i - j + k$ and $c = i + j - k$ . A vector in the plane of $a$ and $b$ , whose projection on $c$ is $\frac{1}{\sqrt{3}}$ is