88180 Let \(u\) and \(v\) be non-collinear vectors in \(R^2\). Let \(w\) be the orthogonal projection vector of \(u\) on \(v\). Consider two statements:
(i) Any vector in \(R^2\) can be written as a linear combination of \(u\) and \(v\)
(ii) \(w\) can be written as a linear combination of \(u\) and \(\mathrm{v}\) as \(\mathrm{w}=\mathrm{au}+\mathrm{bv}\) where both \(\mathrm{a}\) and \(\mathrm{b}\) are non zero real numbers.

88181 Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) are 3 non zero vectors such that no two of these are collinear. If vector \(\vec{a}+2 \vec{b}\) is collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathbf{b}}+\mathbf{3} \overrightarrow{\mathbf{c}}\) is collinear with \(\overrightarrow{\mathbf{a}}(\boldsymbol{\lambda}\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals

88183 If the four points with position vectors \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are coplanar, then \(\lambda\) is equal to

88184 Let \(u, v\) and \(w\) be three vectors in \(R\). Then, any vector \(Z \in R^3\) can be written as \(z=a u=b v+c w\) for some scalars \(a, b\) and \(c\) if and only if

88180 Let \(u\) and \(v\) be non-collinear vectors in \(R^2\). Let \(w\) be the orthogonal projection vector of \(u\) on \(v\). Consider two statements:
(i) Any vector in \(R^2\) can be written as a linear combination of \(u\) and \(v\)
(ii) \(w\) can be written as a linear combination of \(u\) and \(\mathrm{v}\) as \(\mathrm{w}=\mathrm{au}+\mathrm{bv}\) where both \(\mathrm{a}\) and \(\mathrm{b}\) are non zero real numbers.

88181 Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) are 3 non zero vectors such that no two of these are collinear. If vector \(\vec{a}+2 \vec{b}\) is collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathbf{b}}+\mathbf{3} \overrightarrow{\mathbf{c}}\) is collinear with \(\overrightarrow{\mathbf{a}}(\boldsymbol{\lambda}\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals

88183 If the four points with position vectors \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are coplanar, then \(\lambda\) is equal to

88184 Let \(u, v\) and \(w\) be three vectors in \(R\). Then, any vector \(Z \in R^3\) can be written as \(z=a u=b v+c w\) for some scalars \(a, b\) and \(c\) if and only if

88180 Let \(u\) and \(v\) be non-collinear vectors in \(R^2\). Let \(w\) be the orthogonal projection vector of \(u\) on \(v\). Consider two statements:
(i) Any vector in \(R^2\) can be written as a linear combination of \(u\) and \(v\)
(ii) \(w\) can be written as a linear combination of \(u\) and \(\mathrm{v}\) as \(\mathrm{w}=\mathrm{au}+\mathrm{bv}\) where both \(\mathrm{a}\) and \(\mathrm{b}\) are non zero real numbers.

88181 Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) are 3 non zero vectors such that no two of these are collinear. If vector \(\vec{a}+2 \vec{b}\) is collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathbf{b}}+\mathbf{3} \overrightarrow{\mathbf{c}}\) is collinear with \(\overrightarrow{\mathbf{a}}(\boldsymbol{\lambda}\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals

88183 If the four points with position vectors \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are coplanar, then \(\lambda\) is equal to

88184 Let \(u, v\) and \(w\) be three vectors in \(R\). Then, any vector \(Z \in R^3\) can be written as \(z=a u=b v+c w\) for some scalars \(a, b\) and \(c\) if and only if

88180 Let \(u\) and \(v\) be non-collinear vectors in \(R^2\). Let \(w\) be the orthogonal projection vector of \(u\) on \(v\). Consider two statements:
(i) Any vector in \(R^2\) can be written as a linear combination of \(u\) and \(v\)
(ii) \(w\) can be written as a linear combination of \(u\) and \(\mathrm{v}\) as \(\mathrm{w}=\mathrm{au}+\mathrm{bv}\) where both \(\mathrm{a}\) and \(\mathrm{b}\) are non zero real numbers.

88181 Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) are 3 non zero vectors such that no two of these are collinear. If vector \(\vec{a}+2 \vec{b}\) is collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathbf{b}}+\mathbf{3} \overrightarrow{\mathbf{c}}\) is collinear with \(\overrightarrow{\mathbf{a}}(\boldsymbol{\lambda}\) being some non-zero scalar) then \(\vec{a}+2 \vec{b}+6 \vec{c}\) equals

88183 If the four points with position vectors \(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are coplanar, then \(\lambda\) is equal to

88184 Let \(u, v\) and \(w\) be three vectors in \(R\). Then, any vector \(Z \in R^3\) can be written as \(z=a u=b v+c w\) for some scalars \(a, b\) and \(c\) if and only if