Advertisements
Advertisements
प्रश्न
ABCDE is a pentagon, prove that
\[\overrightarrow{AB} + \overrightarrow{AE} + \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{ED} + \overrightarrow{AC} = 3\overrightarrow{AC}\]
उत्तर
Given: ABCDE is a pentagon.
To Prove: \[\overrightarrow{AB} + \overrightarrow{AE} + \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{ED} + \overrightarrow{AC} = 3 \overrightarrow{AC} .\]
Proof: We have,
\[LHS = \overrightarrow{AB} + \overrightarrow{AE} + \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{ED} + \overrightarrow{AC} \]
\[= \left( \overrightarrow{AB} + \overrightarrow{BC} \right) + \left( \overrightarrow{AE} + \overrightarrow{ED} \right) + \overrightarrow{DC} + \overrightarrow{AC}\]
\[= \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{DC} + \overrightarrow{AC}\] [∵ \[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\] and \[\overrightarrow{AE} + \overrightarrow{ED} = \overrightarrow{AD}\]]
\[\overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AC} - \overrightarrow{AD} + \overrightarrow{AC}\] [∵ \[\overrightarrow{AD} + \overrightarrow{DC} = \overrightarrow{AC} \Rightarrow \overrightarrow{DC} = \overrightarrow{AC} - \overrightarrow{AD} \]]
\[= 3 \overrightarrow{AC} = RHS\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the values of x and y so that the vectors `2hati + 3hatj and xhati + yhatj` are equal.
Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).
Find the sum of the vectors `veca = hati -2hatj + hatk, vecb = -2hati + 4hatj + 5hatk and vecc = hati - 6hatj - 7hatk.`
If `veca` and `vecb` are two collinear vectors, then which of the following are incorrect:
A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
If `veca = vecb + vecc`, then is it true that `|veca| = |vecb| + |vecc|`? Justify your answer.
Let `veca = hati + 4hatj + 2hatk, vecb = 3hati - 2hatj + 7hatk ` and `vecc = 2hati - hatj + 4hatk`. Find a vector `vecd` which is perpendicular to both `veca` and `vecb`, and `vecc.vecd = 15`.
ABCD is a quadrilateral. Find the sum the vectors \[\overrightarrow{BA} , \overrightarrow{BC} , \overrightarrow{CD}\] and \[\overrightarrow{DA}\]
ABCDE is a pentagon, prove that
\[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} + \overrightarrow{EA} = \overrightarrow{0}\]
Prove that the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector.
If P is a point and ABCD is a quadrilateral and \[\overrightarrow{AP} + \overrightarrow{PB} + \overrightarrow{PD} = \overrightarrow{PC}\], show that ABCD is a parallelogram.
ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD; BC and AD. Show that\[\vec{PA} + \vec{PB} + \vec{PC} + \vec{PD} = 4 \vec{PQ}\], where P is any point.
Prove that the points \[\hat{i} - \hat{j} , 4 \hat{i} + 3 \hat{j} + \hat{k} \text{ and }2 \hat{i} - 4 \hat{j} + 5 \hat{k}\] are the vertices of a right-angled triangle.
Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.
If `6hati + 10hatj + 3hatk = x(hati + 3hatj + 5hatk) + y(hati - hatj + 5hatk) + z(hati + 3hatj - 4hatk)`, then ______
Find the value of λ such that the vectors `vec"a" = 2hat"i" + lambdahat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"` are orthogonal ______.
Let the position vectors of the points A, Band C be `veca, vecb` and `vecc` respectively. Let Q be the point of intersection of the medians of the triangle ΔABC. Then `vec(QA) + vec(QB) + vec(QC)` =
A vector whose initial and terminal point continues is known as:-
Find the value of `x` and `y`. so that the vectors `2hatj + 3hatj` and `xhati + yhati` are equal
If in ΔABC, `vec(BA) = 2veca` and `vec(BC) = 3vecb`, then `vec(AC)` is ______.
ABCD is a rhombus whose diagonals intersect at E . Then `vec(EA) + vec(EB) + vec(EC) + vec(ED)` equals to ______.
