Advertisements
Advertisements
Question
In the given figure express `bar"c"` and `bar"d"` in terms of `bar"a"` and `bar"b"`.
Solution
`bar"PQ" = bar"PS" + bar"SQ"`
∴ `bar"a" = bar"c" - bar"d"` ...(1)
`bar"PR" = bar"PS" + bar"SR"`
∴ `bar"b" = bar"c" + bar"d"` ....(2)
Adding equations (1) and (2), we get,
`bar"a" + bar"b" = (bar"c" - bar"d") + (bar"c" + bar"d") = 2bar"c"`
∴ `bar"c" = 1/2 (bar"a" + bar"b")`
∴ `bar"c" = 1/2 bar"a" + 1/2bar"b"`
Substituting for `bar"c"` in (2), we get,
∴ `bar"b" = bar"c" + bar"d"`
∴ `bar"d" = bar"b" − bar"c"`
∴ `bar"d" = bar"b" − 1/2 (bar"a" + bar"b")`
∴ `bar"d" = (2bar"b" − (bar"a" + bar"b"))/2`
∴ `bar"d" = (2bar"b" − bar"a" − bar"b")/2`
∴ `bar"d" = (bar"b" − bar"a")/2`
∴ `bar"d" = 1/2 (bar"b" − bar"a")`
∴ `bar"d" = 1/2bar"b" - 1/2bar"a"`
Hence, `bar"c" = 1/2 bar"a" + 1/2bar"b"` and `bar"d" = 1/2bar"b" - 1/2bar"a"`.
APPEARS IN
RELATED QUESTIONS
If `veca=xhati+2hatj-zhatk and vecb=3hati-yhatj+hatk` are two equal vectors ,then write the value of x+y+z
If G denotes the centroid of ∆ABC, then write the value of \[\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} .\]
Write a unit vector making equal acute angles with the coordinates axes.
Write a unit vector in the direction of \[\overrightarrow{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .\]
Find a unit vector in the direction of the vector \[\overrightarrow{a} = 3 \hat{i} - 2 \hat{j} + 6 \hat{k}\].
If \[\left| \overrightarrow{a} \right| = 4\] and \[- 3 \leq \lambda \leq 2\], then write the range of \[\left| \lambda \vec{a} \right|\].
If O and O' are circumcentre and orthocentre of ∆ ABC, then \[\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC}\] equals
If \[\vec{a}\text{ and }\vec{b}\] are two collinear vectors, then which of the following are incorrect?
Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line \[\frac{x + 2}{5} = \frac{y - 3}{4} = \frac{z}{5}\] or not.
Find the components along the coordinate axes of the position vector of the following point :
P(3, 2)
Find the position vector of the mid-point of the vector joining the points
Find the value of λ for which the four points with position vectors `6hat"i" - 7hat"j", 16hat"i" - 19hat"j" - 4hat"k" , lambdahat"j" - 6hat"k" "and" 2hat"i" - 5hat"j" + 10hat"k"` are coplanar.
Find the coordinates of the point which is located three units behind the YZ-plane, four units to the right of XZ-plane, and five units above the XY-plane.
Select the correct option from the given alternatives:
If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______
Let `bara = hati - hatj, barb = hatj - hatk, barc = hatk - hati.` If `bard` is a unit vector such that `bara * bard = 0 = [(barb, barc, bard)]`, then `bard` equals ______.
Dot product of a vector with vectors `3hat"i" - 5hat"k", 2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k"` are respectively -1, 6 and 5. Find the vector.
Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.
Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`bar"a" xx(bar"b" xx bar"c")`
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`(bar"a".bar"b").bar"c"`
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`(bar"a".bar"b")bar"c"`
State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:
`|bar"a"|. (bar"b" + bar"c")`
For 0 < θ < π, if A = `[(costheta, -sintheta), (sintheta, costheta)]`, then ______
lf `overlinea`, `overlineb` and `overlinec` are unit vectors such that `overlinea + overlineb + overlinec = overline0` and angle between `overlinea` and `overlineb` is `pi/3`, then `|overlinea xx overlineb| + |overlineb xx overlinec| + |overlinec xx overlinea|` = ______
The vector with initial point P (2, –3, 5) and terminal point Q(3, –4, 7) is ______.
If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`
Find a unit vector in the direction of `vec"PQ"`, where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively
The formula `(vec"a" + vec"b")^2 = vec"a"^2 + vec"b"^2 + 2vec"a" xx vec"b"` is valid for non-zero vectors `vec"a"` and `vec"b"`
Classify the following measures as scalar and vector.
20 m/s2
Classify the following as scalar and vector quantity.
Velocity
Classify the following as scalar and vector quantity.
Work done
In Figure, identify the following vector.
Collinear but not equal
The unit vector perpendicular to the vectors `6hati + 2hatj + 3hatk` and `3hati - 6hatj - 2hatk` is
Find `|veca xx vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk` and `3hati + 4hatk` form a triangle or not.
Evaluate the following.
`int x^3/(sqrt1 + x^4) `dx
Check whether the vectors `2hati + 2hatj +3hatk, - 3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
