If O is a Point in Space, Abc is a Triangle and D, E, F Are the Mid-points of the Sides Bc, Ca and Ab Respectively of the Triangle, Prove that \[\Vec{Oa} + - Mathematics

प्रश्न

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that $\vec{O A} + \vec{O B} + \vec{O C} = \vec{O D} + \vec{O E} + \vec{O F}$

संक्षेप में उत्तर

उत्तर

Let D, E and F are the midpoints of BC, CA and AB respectively.
Therefore,
$\frac{\vec{O B} + \vec{O C}}{2} = \vec{O D}$
$\vec{O B} + \vec{O C} = 2 \vec{O D} . . . . . (1)$
Similarly,
$\vec{O C} + \vec{O A} = 2 \vec{O E} . . . . . (2)$
$\vec{O A} + \vec{O B} = 2 \vec{O F} . . . . . (3)$
Adding (1), (2) and (3). We get,

$2 (\vec{O A} + \vec{O B} + \vec{O C}) = 2 (\vec{O D} + \vec{O E} + \vec{O F}) .$

$\Rightarrow \vec{O A} + \vec{O B} + \vec{O C} = \vec{O D} + \vec{O E} + \vec{O F} .$

Hence Proved.

shaalaa.com

Introduction of Vector

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 1 Q 6 Q 2

अध्याय 23: Algebra of Vectors - Exercise 23.4 [पृष्ठ ३६]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 23 Algebra of Vectors
Exercise 23.4 | Q 1 | पृष्ठ ३६

वीडियो ट्यूटोरियलVIEW ALL [2]

view
वीडियो ट्यूटोरियल For All Subjects
Introduction of Vector

video tutorial00:52:01
Introduction of Vector

video tutorial00:19:49

संबंधित प्रश्न

Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec²

Answer the following as true or false:
$\vec{a}$ and $\vec{a}$ are collinear.

Answer the following as true or false:
Two collinear vectors are always equal in magnitude.

Answer the following as true or false:
Two vectors having same magnitude are collinear.

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{a}$ + $\vec{b}$ and $\vec{a}$ − $\vec{b}$ .

If $\vec{a}$ is a vector and m is a scalar such that m $\vec{a}$ = $\vec{0}$ , then what are the alternatives for m and $\vec{a}$ ?

If the vectors $\vec{a} = 2 \hat{i} - 3 \hat{j}$ and $\vec{b} = - 6 \hat{i} + m \hat{j}$ are collinear, find the value of m.

If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(1) $5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c} and 3 \vec{a} + 20 \vec{b} + 5 \vec{c}$

(2)

\vec{a} - 2 \vec{b} + 3 \vec{c}, - 3 \vec{b} + 5 \vec{c} and - 2 \vec{a} + 3 \vec{b} - 4 \vec{c}

Prove that the following vectors are coplanar:
$2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k} and 3 \hat{i} - 4 \hat{j} - 4 \hat{k}$

Prove that the following vectors are coplanar:
$\hat{i} + \hat{j} + \hat{k}, 2 \hat{i} + 3 \hat{j} - \hat{k} and - \hat{i} - 2 \hat{j} + 2 \hat{k}$

If $\vec{a} \cdot i = \vec{a} \cdot (\hat{i} + \hat{j}) = \vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1,$ then $\vec{a} =$

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}, | \vec{a} | = 3, | \vec{b} | = 5, | \vec{c} | = 7,$ then the angle between $\vec{a} and \vec{b}$ is

Let $\vec{a} and \vec{b}$ be two unit vectors and α be the angle between them. Then, $\vec{a} + \vec{b}$ is a unit vector if

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}, | \vec{a} | = 3, | \vec{b} | = 5, | \vec{c} | = 7,$ then the angle between $\vec{a} and \vec{b}$ is

The vector (cos α cos β) $\hat{i}$ + (cos α sin β) $\hat{j}$ + (sin α) $\hat{k}$ is a

If the position vectors of P and Q are $\hat{i} + 3 \hat{j} - 7 \hat{k} and 5 i - 2 \hat{j} + 4 \hat{k}$ then the cosine of the angle between $\vec{P Q}$ and y-axis is

If $\vec{a} and \vec{b}$ are unit vectors, then which of the following values of $\vec{a} . \vec{b}$ is not possible?

The vector component of $\vec{b}$ perpendicular to $\vec{a}$ is

What is the length of the longer diagonal of the parallelogram constructed on $5 \vec{a} + 2 \vec{b} and \vec{a} - 3 \vec{b}$ if it is given that $| \vec{a} | = 2 \sqrt{2}, | \vec{b} | = 3$ and the angle between $\vec{a} and \vec{b}$ is π/4?

If $\vec{a}$ is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ $\vec{a}$ is a unit vector if

If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then, $\vec{a} \cdot \vec{b}$ ≥ 0, only when

The values of x for which the angle between $\vec{a} = 2 x^{2} \hat{i} + 4 x \hat{j} + \hat{k}, \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$ is obtuse and the angle between $\vec{b}$ and the z-axis is acute and less than $\frac{π}{6}$ are

If $\vec{a}, \vec{b}, \vec{c}$ are any three mutually perpendicular vectors of equal magnitude a, then $| \vec{a} + \vec{b} + \vec{c} |$ is equal to

If the vectors $3 \hat{i} + λ \hat{j} + \hat{k} and 2 \hat{i} - \hat{j} + 8 \hat{k}$ are perpendicular, then λ is equal to

If $\vec{a} and \vec{b}$ are unit vectors inclined at an angle θ, then the value of $| \vec{a} - \vec{b} |$

If $\vec{a} and \vec{b}$ are unit vectors, then the greatest value of $\sqrt{3} | \vec{a} + \vec{b} | + | \vec{a} - \vec{b} |$

Let $\vec{a}, \vec{b}, \vec{c}$ be three unit vectors, such that $| \vec{a} + \vec{b} + \vec{c} |$ =1 and $\vec{a}$ is perpendicular to $\vec{b}$ If $\vec{c}$ makes angles α and β with $\vec{a} a n d \vec{b}$ respectively, then cos α + cos β =

If $\vec{a} and \vec{b}$ be two unit vectors and θ the angle between them, then $\vec{a} + \vec{b}$ is a unit vector if θ =

If O is a Point in Space, Abc is a Triangle and D, E, F Are the Mid-points of the Sides Bc, Ca and Ab Respectively of the Triangle, Prove that \[\Vec{Oa} + - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्न

Select a course

If O is a Point in Space, Abc is a Triangle and D, E, F Are the Mid-points of the Sides Bc, Ca and Ab Respectively of the Triangle, Prove that \[\Vec{Oa} + - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्न

Select a course

उत्तर