87766 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathbf{c}}\) are three vectors of which every pair is non-collinear. If the vector \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) are collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathrm{a}}\) respectively, then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\) is
87767 Let \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{b}\) three con-coplanar vectors, and let \(\vec{p}, \vec{q}\) are \(\overrightarrow{\mathbf{r}}\) be vectors defined by the relations \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a b c}}]}, \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a b c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a b c}}]}\) then the value of the expression \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{q}}+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{r}}\) is equal to
87769 Let us define the length of a vector \(2 \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}=\mathbf{a}+\mathbf{b}+\mathbf{c}\). The definition concides with the usual definition of length of a vector \(\mathbf{a i}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c k}\) iff
87770 If \(\hat{a}, \hat{b}, \hat{c}\) are three unit vectors such that \(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then what is the value of \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}\) ?
87766 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathbf{c}}\) are three vectors of which every pair is non-collinear. If the vector \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) are collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathrm{a}}\) respectively, then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\) is
87767 Let \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{b}\) three con-coplanar vectors, and let \(\vec{p}, \vec{q}\) are \(\overrightarrow{\mathbf{r}}\) be vectors defined by the relations \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a b c}}]}, \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a b c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a b c}}]}\) then the value of the expression \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{q}}+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{r}}\) is equal to
87769 Let us define the length of a vector \(2 \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}=\mathbf{a}+\mathbf{b}+\mathbf{c}\). The definition concides with the usual definition of length of a vector \(\mathbf{a i}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c k}\) iff
87770 If \(\hat{a}, \hat{b}, \hat{c}\) are three unit vectors such that \(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then what is the value of \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}\) ?
87766 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathbf{c}}\) are three vectors of which every pair is non-collinear. If the vector \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) are collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathrm{a}}\) respectively, then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\) is
87767 Let \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{b}\) three con-coplanar vectors, and let \(\vec{p}, \vec{q}\) are \(\overrightarrow{\mathbf{r}}\) be vectors defined by the relations \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a b c}}]}, \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a b c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a b c}}]}\) then the value of the expression \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{q}}+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{r}}\) is equal to
87769 Let us define the length of a vector \(2 \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}=\mathbf{a}+\mathbf{b}+\mathbf{c}\). The definition concides with the usual definition of length of a vector \(\mathbf{a i}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c k}\) iff
87770 If \(\hat{a}, \hat{b}, \hat{c}\) are three unit vectors such that \(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then what is the value of \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}\) ?
87766 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathbf{c}}\) are three vectors of which every pair is non-collinear. If the vector \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) are collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathrm{a}}\) respectively, then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\) is
87767 Let \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{b}\) three con-coplanar vectors, and let \(\vec{p}, \vec{q}\) are \(\overrightarrow{\mathbf{r}}\) be vectors defined by the relations \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a b c}}]}, \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a b c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a b c}}]}\) then the value of the expression \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{q}}+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{r}}\) is equal to
87769 Let us define the length of a vector \(2 \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}=\mathbf{a}+\mathbf{b}+\mathbf{c}\). The definition concides with the usual definition of length of a vector \(\mathbf{a i}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c k}\) iff
87770 If \(\hat{a}, \hat{b}, \hat{c}\) are three unit vectors such that \(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then what is the value of \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}\) ?
87766 \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathbf{c}}\) are three vectors of which every pair is non-collinear. If the vector \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) are collinear with \(\overrightarrow{\mathbf{c}}\) and \(\overrightarrow{\mathrm{a}}\) respectively, then \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}\) is
87767 Let \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{b}\) three con-coplanar vectors, and let \(\vec{p}, \vec{q}\) are \(\overrightarrow{\mathbf{r}}\) be vectors defined by the relations \(\overrightarrow{\mathbf{p}}=\frac{\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}}{[\overrightarrow{\mathbf{a b c}}]}, \overrightarrow{\mathbf{q}}=\frac{\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}}{[\overrightarrow{\mathbf{a b c}}]}\) and \(\overrightarrow{\mathbf{r}}=\frac{\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}}{[\overrightarrow{\mathbf{a b c}}]}\) then the value of the expression \((\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{p}}+(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \overrightarrow{\mathbf{q}}+(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{r}}\) is equal to
87769 Let us define the length of a vector \(2 \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c} \hat{\mathbf{k}}=\mathbf{a}+\mathbf{b}+\mathbf{c}\). The definition concides with the usual definition of length of a vector \(\mathbf{a i}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{c k}\) iff
87770 If \(\hat{a}, \hat{b}, \hat{c}\) are three unit vectors such that \(\hat{\mathbf{a}}+\hat{\mathbf{b}}+\hat{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then what is the value of \(\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}+\hat{\mathbf{b}} \cdot \hat{\mathbf{c}}+\hat{\mathbf{c}} \cdot \hat{\mathbf{a}}\) ?