Class 12

Math

Algebra

Vector Algebra

यदि $a=b+c$, तब क्या यह सत्य है कि | $∣a∣=∣∣ b∣∣ +∣c∣$ ? अपने उत्तर की पुष्टि कीजिए।

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If $D,EandF$ are three points on the sides $BC,CAandAB,$ respectively, of a triangle $ABC$ such that the $CDBD ,AECE ,BFAF =−1$

Points $A(a),B(b),C(c)andD(d)$ are relates as $xa+yb+zc+wd=0$ and $x+y+z+w=0,wherex,y,z,andw$ are scalars (sum of any two of $x,y,znadw$ is not zero). Prove that if $A,B,CandD$ are concylic, then $∣xy∣∣∣ a−b∣∣ _{2}=∣wz∣∣∣ c−d∣∣ _{2}˙$

If $i^−3j^ +5k^$ bisects the angle between $a^and−i^+2j^ +2k^,wherea^$ is a unit vector, then a. $a^=1051 (41i^+88j^ −40k^)$ b. $a^=1051 (41i^+88j^ +40k^)$ c. $a^=1051 (−41i^+88j^ −40k^)$ d. $a^=1051 (41i^−88j^ −40k^)$

If $a×b=b×c=0,wherea,b,andc$ are coplanar vectors, then for some scalar $k$ prove that $a+c=kb˙$

A ship is sailing towards the north at a speed of 1.25 m/s. The current is taking it towards the east at the rate of 1 m/s and a sailor is climbing a vertical pole on the ship at the rate of 0.5 m/s. Find the velocity of the sailor in space.

Vectors $a=i^+2j^ +3k^,b=2i^−j^ +k^$ and $c=3i^+j^ +4k^,$ are so placed that the end point of one vector is the starting point of the next vector. Then the vector are (A) not coplanar (B) coplanar but cannot form a triangle (C) coplanar and form a triangle (D) coplanar and can form a right angled triangle

If the vectors $A,B,C$ of a triangle $ABC$ are $(1,2,3),(−1,0,0),(0,1,2),$ respectively then find $∠ABC˙$

In triangle $ABC,∠A=30_{0},H$ is the orthocenter and $D$ is the midpoint of $BC$. Segment $HD$ is produced to $T$ such that $HD=DT$ The length $AT$ is equal to (a). $2BC$ (b). $3BC$ (c). $24 BC$ (d). none of these