Class 12

Math

Algebra

Vector Algebra

Write two different vectors having same magnitude.

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If $a,b,c$ are non-coplanar vector and $λ$ is a real number, then the vectors $a+2b+3c,λb+μcand(2λ−1)c$ are coplanar when a. $μ∈R$ b. $λ=21 $ c. $λ=0$ d. no value of $λ$

If the resultant of three forces $F_{1}=pi^+3j^ −k^,F_{2}=6i^−k^andF_{3}=−5i^+j^ +2k^$ acting on a parricle has magnitude equal to 5 units, then the value of $p$ is a. $−6$ b. $−4$ c. $2$ d. $4$

Prove that vectors $u=(al+a_{1}l_{1})i^+(am+a_{1}m_{1})j^ +(an+a_{1}n_{1})k^$ $v=(bl+b_{1}l_{1})i^+(bm+b_{1}m_{1})j^ +(bn+b_{1}n_{1})k^$ $w=(bl+b_{1}l_{1})i^+(bm+b_{1}m_{1})j^ +(bn+b_{1}n_{1})k^$ are coplanar.

If $a,bandc$ are non-cop0lanar vector, then that prove $∣∣ (ad˙)(b×c)+(bd˙)(c×a)+(cd˙)(a×b)∣∣ $ is independent of $d,wheree$ is a unit vector.

If $a=7i^−4k^andb=−2i^−j^ +2k^,$ determine vector $c$ along the internal bisector of the angle between of the angle between vectors $aandbsuchthat∣c∣$ =5$6 $

Check whether the three vectors $2i^+2j^ +3k^,−3i^+3j^ +2k^and3i^+4k^$ from a triangle or not

If $a,b$ are two non-collinear vectors, prove that the points with position vectors $a+b,a−b$ and $a+λb$ are collinear for all real values of $λ˙$

Show that $(a−b)×(a+b)=2a×b$ and given a geometrical interpretation of it.