Assertion : If a =3 i + k , ~b =- i +2 j + k , c = i + j + k and d =2 i - j , then there…

Assertion : If \(\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{~b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\)and \(\overrightarrow{\mathrm{d}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}\), then there exist real numbers α, β, γ such that \(\overrightarrow{\mathrm{a}}=\alpha \overrightarrow{\mathrm{b}}+\beta \overrightarrow{\mathrm{c}}+\gamma \overrightarrow{\mathrm{d}}\)
Reason : \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{~b}}, \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{~d}}\) are four vectors in a 3–dimensional space. If \(\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{~d}}\) are non–coplanar, then there exist real numbers α, β, γ such that \(\overrightarrow{\mathrm{a}}=\alpha \overrightarrow{\mathrm{b}}+\beta \overrightarrow{\mathrm{c}}+\gamma \overrightarrow{\mathrm{d}}\)