If `l_1^2+m_1^2+n_1^2=1` etc., and `l_1 l_2+m_1 m_2+n_1 n_2 = 0`, etc. and `Delta=|(l_1,m_1,n_1),(l_2,m_2,n_2),(l_3,m_3,n_3)|` then `|Delta|=?`

If l,m and n are real numbers such that l^2+m^2+n^2=0 , then show that |{:(1+l^2,lm,ln),(lm,1+m^2,mn),(ln,mn,1+n^2):}|=1

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 are proportional to l_1+l_2,m_1+m_2, n_1+n_2dot Statement 2: The angle between the two intersection lines having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 is given by costheta=l_1l_2+m_1m_2+n_1n_2dot

The combined equation of the lines l_1a n dl_2 is 2x^2+6x y+y^2=0 and that of the lines m_1a n dm_2 is 4x^2+18 x y+y^2=0 . If the angle between l_1 and m_2 is alpha then the angle between l_2a n dm_1 will be

If m = sqrtfrac(n)(n+1/2) and m = 1/2, then n =

A B C is a right-angled triangle in which /_B=90^0 and B C=adot If n points L_1, L_2, ,L_nonA B is divided in n+1 equal parts and L_1M_1, L_2M_2, ,L_n M_n are line segments parallel to B Ca n dM_1, M_2, ,M_n are on A C , then the sum of the lengths of L_1M_1, L_2M_2, ,L_n M_n is (a(n+1))/2 b. (a(n-1))/2 c. (a n)/2 d. none of these

Prove that vectors vec u=(a l+a_1l_1) hat i+(a m+a_1m_1) hat j+(a n+a_1n_1) hat k vec v=(b l+b_1l_1) hat i+(b m+b_1m_1) hat j+(b n+b_1n_1) hat k vec w=(c l+c_1l_1) hat i+(c m+c_1m_1) hat j+(c n+c_1n_1) hat k are coplanar.

Let A D be a median of the A B Cdot If A Ea n dA F are medians of the triangle A B Da n dA D C , respectively, and A D=m_1,A E=m_2,A F=m_3, then (a^2)/8 is equal to (a) m_2^2+m_3^2-2m_1^2 (b) m_1^2+m_2^2-2m_3^2 (c) m_1^2+m_3^2-2m_2^2 (d) none of these

What is shape of the orbital with (i) n = 2 and l = 0 , (ii) n = 2 and l = 1?

ABCD is a square of length a, a in N , a > 1. Let L_1, L_2 , L_3... be points on BC such that BL_1 = L_1 L_2 = L_2 L_3 = .... 1 and M_1,M_2 , M_3,.... be points on CD such that CM_1 = M_1M_2= M_2 M_3=... = 1 . Then sum_(n = 1)^(a-1) ((AL_n)^2 + (L_n M_n)^2) is equal to :